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Metamath Proof Explorer

Theorem r19.21v

Description: Restricted quantifier version of 19.21v . (Contributed by NM, 15-Oct-2003) (Proof shortened by Andrew Salmon, 30-May-2011) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020) (Proof shortened by Wolf Lammen, 11-Dec-2024)

Ref Expression
Assertion r19.21v 
|- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )

Proof

Step Hyp Ref Expression
1 pm2.27 
 |-  ( ph -> ( ( ph -> ps ) -> ps ) )
2 1 ralimdv 
 |-  ( ph -> ( A. x e. A ( ph -> ps ) -> A. x e. A ps ) )
3 2 com12 
 |-  ( A. x e. A ( ph -> ps ) -> ( ph -> A. x e. A ps ) )
4 pm2.21 
 |-  ( -. ph -> ( ph -> ps ) )
5 4 ralrimivw 
 |-  ( -. ph -> A. x e. A ( ph -> ps ) )
6 ax-1 
 |-  ( ps -> ( ph -> ps ) )
7 6 ralimi 
 |-  ( A. x e. A ps -> A. x e. A ( ph -> ps ) )
8 5 7 ja 
 |-  ( ( ph -> A. x e. A ps ) -> A. x e. A ( ph -> ps ) )
9 3 8 impbii 
 |-  ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )