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Metamath Proof Explorer

Theorem bj-pm11.53a

Description: A variant of pm11.53v . One can similarly prove a variant with DV ( y , ph ) and A. y F// x ps instead of DV ( x , ps ) and A. x F// y ph . (Contributed by BJ, 7-Oct-2024)

		Ref	Expression
	Assertion	bj-pm11.53a	\|- ( A. x F// y ph -> ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-nnfv	\|- F// x A. y ps
2		bj-pm11.53vw	\|- ( ( A. x F// y ph /\ F// x A. y ps ) -> ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) ) )
3	1 2	mpan2	\|- ( A. x F// y ph -> ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) ) )