Theorem bj-nnflemae

Description: One of four lemmas for nonfreeness: antecedent expressed with universal quantifier and consequent expressed with existential quantifier. (Contributed by BJ, 12-Aug-2023) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-nnflemae	|- ( A. x ( ph -> A. y ph ) -> ( E. x ph -> A. y E. x ph ) )

Proof

Step	Hyp	Ref	Expression
1		exim	|- ( A. x ( ph -> A. y ph ) -> ( E. x ph -> E. x A. y ph ) )
2		bj-19.12	|- ( E. x A. y ph -> A. y E. x ph )
3	1 2	syl6	|- ( A. x ( ph -> A. y ph ) -> ( E. x ph -> A. y E. x ph ) )