Metamath Proof Explorer

Theorem r19.23t

Description: Closed theorem form of r19.23 . (Contributed by NM, 4-Mar-2013) (Revised by Mario Carneiro, 8-Oct-2016)

Ref Expression
Assertion r19.23t 
|- ( F/ x ps -> ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) )

Proof

Step Hyp Ref Expression
1 19.23t 
 |-  ( F/ x ps -> ( A. x ( ( x e. A /\ ph ) -> ps ) <-> ( E. x ( x e. A /\ ph ) -> ps ) ) )
2 df-ral 
 |-  ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
3 impexp 
 |-  ( ( ( x e. A /\ ph ) -> ps ) <-> ( x e. A -> ( ph -> ps ) ) )
4 3 albii 
 |-  ( A. x ( ( x e. A /\ ph ) -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
5 2 4 bitr4i 
 |-  ( A. x e. A ( ph -> ps ) <-> A. x ( ( x e. A /\ ph ) -> ps ) )
6 df-rex 
 |-  ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
7 6 imbi1i 
 |-  ( ( E. x e. A ph -> ps ) <-> ( E. x ( x e. A /\ ph ) -> ps ) )
8 1 5 7 3bitr4g 
 |-  ( F/ x ps -> ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) )