Theorem rexeqbi1dv

Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997) (Proof shortened by Steven Nguyen, 5-May-2023)

	Ref	Expression
Hypothesis	raleqbi1dv.1	|- ( A = B -> ( ph <-> ps ) )
Assertion	rexeqbi1dv	|- ( A = B -> ( E. x e. A ph <-> E. x e. B ps ) )

Proof

Step	Hyp	Ref	Expression
1		raleqbi1dv.1	|- ( A = B -> ( ph <-> ps ) )
2		id	|- ( A = B -> A = B )
3	2 1	rexeqbidv	|- ( A = B -> ( E. x e. A ph <-> E. x e. B ps ) )