Theorem rmoeqd

Description: Equality deduction for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017)

	Ref	Expression
Hypothesis	raleqd.1	|- ( A = B -> ( ph <-> ps ) )
Assertion	rmoeqd	|- ( 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		rmoeq1	|- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )
3	1	rmobidv	|- ( 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 ) )