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Metamath Proof Explorer

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 ) ) )