diff --git a/src/ast/rewriter/bool_rewriter.cpp b/src/ast/rewriter/bool_rewriter.cpp
index 9afab7a2911..13a392d247d 100644
--- a/src/ast/rewriter/bool_rewriter.cpp
+++ b/src/ast/rewriter/bool_rewriter.cpp
@@ -699,6 +699,22 @@ app* bool_rewriter::mk_eq(expr* lhs, expr* rhs) {
     return m().mk_eq(lhs, rhs);
 }
 
+bool bool_rewriter::try_ite_eq(expr* lhs, expr* rhs, expr_ref& r) {
+    expr* c, *t, *e;
+    if (!m().is_ite(lhs, c, t, e))
+        return false;
+    if (m().are_equal(t, rhs) && m().are_distinct(e, rhs)) {
+        r = c;
+        return true;
+    }
+    if (m().are_equal(e, rhs) && m().are_distinct(t, rhs)) {
+        r = m().mk_not(c);
+        return true;
+    }
+    return false;
+}
+
+
 br_status bool_rewriter::mk_eq_core(expr * lhs, expr * rhs, expr_ref & result) {
     if (m().are_equal(lhs, rhs)) {
         result = m().mk_true();
@@ -713,6 +729,12 @@ br_status bool_rewriter::mk_eq_core(expr * lhs, expr * rhs, expr_ref & result) {
     br_status r = BR_FAILED;
     
 
+    if (try_ite_eq(lhs, rhs, result))
+        return BR_REWRITE1;
+
+    if (try_ite_eq(rhs, lhs, result))
+        return BR_REWRITE1;
+    
     if (m_ite_extra_rules) {
         if (m().is_ite(lhs) && m().is_value(rhs)) {
             r = try_ite_value(to_app(lhs), to_app(rhs), result);
diff --git a/src/ast/rewriter/bool_rewriter.h b/src/ast/rewriter/bool_rewriter.h
index 7c840b6478c..421811ed4d1 100644
--- a/src/ast/rewriter/bool_rewriter.h
+++ b/src/ast/rewriter/bool_rewriter.h
@@ -71,6 +71,8 @@ class bool_rewriter {
 
     void mk_and_as_or(unsigned num_args, expr * const * args, expr_ref & result);
 
+    bool try_ite_eq(expr* lhs, expr* rhs, expr_ref& r);
+
     expr * mk_or_app(unsigned num_args, expr * const * args);
     bool simp_nested_not_or(unsigned num_args, expr * const * args, expr_fast_mark1 & neg_lits, expr_fast_mark2 & pos_lits, expr_ref & result);
     expr * simp_arg(expr * arg, expr_fast_mark1 & neg_lits, expr_fast_mark2 & pos_lits, bool & modified);
diff --git a/src/sat/smt/q_ematch.cpp b/src/sat/smt/q_ematch.cpp
index aa79bab5b41..76d234d8d5a 100644
--- a/src/sat/smt/q_ematch.cpp
+++ b/src/sat/smt/q_ematch.cpp
@@ -390,7 +390,7 @@ namespace q {
         m_qs.log_instantiation(lits, &j);
         euf::th_proof_hint* ph = nullptr;
         if (ctx.use_drat()) 
-            ph = q_proof_hint::mk(ctx, j.m_generation, lits, j.m_clause.num_decls(), j.m_binding);
+            ph = q_proof_hint::mk(ctx, m_ematch, j.m_generation, lits, j.m_clause.num_decls(), j.m_binding);
         m_qs.add_clause(lits, ph);               
     }
 
diff --git a/src/sat/smt/q_ematch.h b/src/sat/smt/q_ematch.h
index cbeb34679bf..f7de55fb813 100644
--- a/src/sat/smt/q_ematch.h
+++ b/src/sat/smt/q_ematch.h
@@ -90,6 +90,7 @@ namespace q {
         unsigned_vector               m_clause_queue;
         euf::enode_pair_vector        m_evidence;
         bool                          m_enable_propagate = true;
+        symbol                        m_ematch = symbol("ematch");
 
         euf::enode* const* copy_nodes(clause& c, euf::enode* const* _binding);
         binding* tmp_binding(clause& c, app* pat, euf::enode* const* _binding);
diff --git a/src/sat/smt/q_mbi.cpp b/src/sat/smt/q_mbi.cpp
index 539c4f943ba..07d4880c966 100644
--- a/src/sat/smt/q_mbi.cpp
+++ b/src/sat/smt/q_mbi.cpp
@@ -71,7 +71,7 @@ namespace q {
         for (auto const& [qlit, fml, inst, generation] : m_instantiations) {
             euf::solver::scoped_generation sg(ctx, generation + 1);
             sat::literal lit = ~ctx.mk_literal(fml);
-            auto* ph = ctx.use_drat()? q_proof_hint::mk(ctx, generation, ~qlit, lit, inst.size(), inst.data()) : nullptr;
+            auto* ph = ctx.use_drat()? q_proof_hint::mk(ctx, m_mbqi, generation, ~qlit, lit, inst.size(), inst.data()) : nullptr;
             m_qs.add_clause(~qlit, lit, ph);
             m_qs.log_instantiation(~qlit, lit);
         }
diff --git a/src/sat/smt/q_mbi.h b/src/sat/smt/q_mbi.h
index 96e3ba56f97..71a15be7473 100644
--- a/src/sat/smt/q_mbi.h
+++ b/src/sat/smt/q_mbi.h
@@ -72,6 +72,7 @@ namespace q {
         unsigned                               m_max_choose_candidates = 10;
         unsigned                               m_generation_bound = UINT_MAX;
         unsigned                               m_generation_max = UINT_MAX;
+        symbol                                 m_mbqi = symbol("mbqi");
         typedef std::tuple<sat::literal, expr_ref, expr_ref_vector, unsigned> instantiation_t;
         vector<instantiation_t> m_instantiations;
         vector<mbp::def>        m_defs;
diff --git a/src/sat/smt/q_solver.cpp b/src/sat/smt/q_solver.cpp
index fff11898c7d..aec10607248 100644
--- a/src/sat/smt/q_solver.cpp
+++ b/src/sat/smt/q_solver.cpp
@@ -364,10 +364,10 @@ namespace q {
         }
     }
 
-    q_proof_hint* q_proof_hint::mk(euf::solver& s, unsigned generation, sat::literal_vector const& lits, unsigned n, euf::enode* const* bindings) {
+    q_proof_hint* q_proof_hint::mk(euf::solver& s, symbol const& method, unsigned generation, sat::literal_vector const& lits, unsigned n, euf::enode* const* bindings) {
         SASSERT(n > 0);
         auto* mem = s.get_region().allocate(q_proof_hint::get_obj_size(n, lits.size()));
-        q_proof_hint* ph = new (mem) q_proof_hint(generation, n, lits.size());
+        q_proof_hint* ph = new (mem) q_proof_hint(method, generation, n, lits.size());
         for (unsigned i = 0; i < n; ++i)
             ph->m_bindings[i] = bindings[i]->get_expr();
         for (unsigned i = 0; i < lits.size(); ++i)
@@ -375,10 +375,10 @@ namespace q {
         return ph;
     }
 
-    q_proof_hint* q_proof_hint::mk(euf::solver& s, unsigned generation, sat::literal l1, sat::literal l2, unsigned n, expr* const* bindings) {
+    q_proof_hint* q_proof_hint::mk(euf::solver& s, symbol const& method, unsigned generation, sat::literal l1, sat::literal l2, unsigned n, expr* const* bindings) {
         SASSERT(n > 0);
         auto* mem = s.get_region().allocate(q_proof_hint::get_obj_size(n, 2));
-        q_proof_hint* ph = new (mem) q_proof_hint(generation, n, 2);
+        q_proof_hint* ph = new (mem) q_proof_hint(method, generation, n, 2);
         for (unsigned i = 0; i < n; ++i)
             ph->m_bindings[i] = bindings[i];
         ph->m_literals[0] = l1;
@@ -402,6 +402,7 @@ namespace q {
             args.push_back(s.literal2expr(~m_literals[i]));
         args.push_back(binding);        
         args.push_back(m.mk_app(symbol("gen"), 1, gens, range));
+        args.push_back(m.mk_const(m_method, range));
         return m.mk_app(symbol("inst"), args.size(), args.data(), range);
     }
 
diff --git a/src/sat/smt/q_solver.h b/src/sat/smt/q_solver.h
index d0581f85203..a7220e68b9d 100644
--- a/src/sat/smt/q_solver.h
+++ b/src/sat/smt/q_solver.h
@@ -30,21 +30,23 @@ namespace euf {
 namespace q {
 
     struct q_proof_hint : public euf::th_proof_hint {
+        symbol        m_method;
         unsigned      m_generation;
         unsigned      m_num_bindings;
         unsigned      m_num_literals;
         sat::literal* m_literals;
         expr*         m_bindings[0];
         
-        q_proof_hint(unsigned g, unsigned b, unsigned l) {
+        q_proof_hint(symbol const& method, unsigned g, unsigned b, unsigned l) {
+            m_method = method;
             m_generation = g;
             m_num_bindings = b;
             m_num_literals = l;
             m_literals = reinterpret_cast<sat::literal*>(m_bindings + m_num_bindings);
         }
         static size_t get_obj_size(unsigned num_bindings, unsigned num_lits) { return sizeof(q_proof_hint) + num_bindings*sizeof(expr*) + num_lits*sizeof(sat::literal); }
-        static q_proof_hint* mk(euf::solver& s, unsigned generation, sat::literal_vector const& lits, unsigned n, euf::enode* const* bindings);
-        static q_proof_hint* mk(euf::solver& s, unsigned generation, sat::literal l1, sat::literal l2, unsigned n, expr* const* bindings);
+        static q_proof_hint* mk(euf::solver& s, symbol const& method, unsigned generation, sat::literal_vector const& lits, unsigned n, euf::enode* const* bindings);
+        static q_proof_hint* mk(euf::solver& s, symbol const& method, unsigned generation, sat::literal l1, sat::literal l2, unsigned n, expr* const* bindings);
         expr* get_hint(euf::solver& s) const override;
     };