Theorem reueqd

Description: Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004)

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

Proof

Step	Hyp	Ref	Expression
1		raleqd.1	|- ( A = B -> ( ph <-> ps ) )
2		reueq1	|- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )
3	1	reubidv	|- ( A = B -> ( E! x e. B ph <-> E! x e. B ps ) )
4	2 3	bitrd	|- ( A = B -> ( E! x e. A ph <-> E! x e. B ps ) )