Theorem bj-19.42t

Description: Closed form of 19.42 from the same axioms as 19.42v . (Contributed by BJ, 2-Dec-2023)

	Ref	Expression
Assertion	bj-19.42t	|- ( F// x ph -> ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		19.40	|- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) )
2		bj-nnfe	|- ( F// x ph -> ( E. x ph -> ph ) )
3	2	anim1d	|- ( F// x ph -> ( ( E. x ph /\ E. x ps ) -> ( ph /\ E. x ps ) ) )
4	1 3	syl5	|- ( F// x ph -> ( E. x ( ph /\ ps ) -> ( ph /\ E. x ps ) ) )
5		bj-nnfa	|- ( F// x ph -> ( ph -> A. x ph ) )
6	5	anim1d	|- ( F// x ph -> ( ( ph /\ E. x ps ) -> ( A. x ph /\ E. x ps ) ) )
7		19.29	|- ( ( A. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) )
8	6 7	syl6	|- ( F// x ph -> ( ( ph /\ E. x ps ) -> E. x ( ph /\ ps ) ) )
9	4 8	impbid	|- ( F// x ph -> ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) ) )