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Metamath Proof Explorer

Theorem r19.35

Description: Restricted quantifier version of 19.35 . (Contributed by NM, 20-Sep-2003) (Proof shortened by Wolf Lammen, 22-Dec-2024)

Ref Expression
Assertion r19.35 
|- ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) )

Proof

Step Hyp Ref Expression
1 pm5.5 
 |-  ( ph -> ( ( ph -> ps ) <-> ps ) )
2 1 ralrexbid 
 |-  ( A. x e. A ph -> ( E. x e. A ( ph -> ps ) <-> E. x e. A ps ) )
3 2 biimpcd 
 |-  ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> E. x e. A ps ) )
4 rexnal 
 |-  ( E. x e. A -. ph <-> -. A. x e. A ph )
5 pm2.21 
 |-  ( -. ph -> ( ph -> ps ) )
6 5 reximi 
 |-  ( E. x e. A -. ph -> E. x e. A ( ph -> ps ) )
7 4 6 sylbir 
 |-  ( -. A. x e. A ph -> E. x e. A ( ph -> ps ) )
8 ax-1 
 |-  ( ps -> ( ph -> ps ) )
9 8 reximi 
 |-  ( E. x e. A ps -> E. x e. A ( ph -> ps ) )
10 7 9 ja 
 |-  ( ( A. x e. A ph -> E. x e. A ps ) -> E. x e. A ( ph -> ps ) )
11 3 10 impbii 
 |-  ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) )