We consider a class of logical formalisms, in which first-order logic is extended by identifying propositions modulo a given congruence. We particularly focus on the case where this congruence is induced by a confluent and terminating rewrite system over the propositions. This extension enhances the power of first-order logic and various formalisms, including higher-order logic, can be described in this framework. We conjecture that proof normalization and logical consistency always hold over this class of formalisms, provided some minimal conditions over the rewrite system are fulfilled. We prove this conjecture for some subcases, including higher-order logic. At last, we extend these results to classical sequent calculus.