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Metamath Proof Explorer

Theorem r19.12sn

Description: Special case of r19.12 where its converse holds. (Contributed by NM, 19-May-2008) (Revised by Mario Carneiro, 23-Apr-2015) (Revised by BJ, 18-Mar-2020)

		Ref	Expression
	Assertion	r19.12sn	\|- ( A e. V -> ( E. x e. { A } A. y e. B ph <-> A. y e. B E. x e. { A } ph ) )

Proof

Step	Hyp	Ref	Expression
1		sbcralg	\|- ( A e. V -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )
2		rexsns	\|- ( E. x e. { A } A. y e. B ph <-> [. A / x ]. A. y e. B ph )
3		rexsns	\|- ( E. x e. { A } ph <-> [. A / x ]. ph )
4	3	ralbii	\|- ( A. y e. B E. x e. { A } ph <-> A. y e. B [. A / x ]. ph )
5	1 2 4	3bitr4g	\|- ( A e. V -> ( E. x e. { A } A. y e. B ph <-> A. y e. B E. x e. { A } ph ) )