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Metamath Proof Explorer

Theorem r19.21t

Description: Restricted quantifier version of 19.21t ; closed form of r19.21 . (Contributed by NM, 1-Mar-2008) (Proof shortened by Wolf Lammen, 2-Jan-2020)

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

Proof

Step Hyp Ref Expression
1 19.21t 
 |-  ( F/ x ph -> ( A. x ( ph -> ( x e. A -> ps ) ) <-> ( ph -> A. x ( x e. A -> ps ) ) ) )
2 df-ral 
 |-  ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
3 bi2.04 
 |-  ( ( x e. A -> ( ph -> ps ) ) <-> ( ph -> ( x e. A -> ps ) ) )
4 3 albii 
 |-  ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ph -> ( x e. A -> ps ) ) )
5 2 4 bitri 
 |-  ( A. x e. A ( ph -> ps ) <-> A. x ( ph -> ( x e. A -> ps ) ) )
6 df-ral 
 |-  ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
7 6 imbi2i 
 |-  ( ( ph -> A. x e. A ps ) <-> ( ph -> A. x ( x e. A -> ps ) ) )
8 1 5 7 3bitr4g 
 |-  ( F/ x ph -> ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) )