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Metamath Proof Explorer

Theorem cbvraldva2

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017)

Ref Expression
Hypotheses cbvraldva2.1 
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
cbvraldva2.2 
|- ( ( ph /\ x = y ) -> A = B )
Assertion cbvraldva2 
|- ( ph -> ( A. x e. A ps <-> A. y e. B ch ) )

Proof

Step Hyp Ref Expression
1 cbvraldva2.1 
 |-  ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2 cbvraldva2.2 
 |-  ( ( ph /\ x = y ) -> A = B )
3 simpr 
 |-  ( ( ph /\ x = y ) -> x = y )
4 3 2 eleq12d 
 |-  ( ( ph /\ x = y ) -> ( x e. A <-> y e. B ) )
5 4 1 imbi12d 
 |-  ( ( ph /\ x = y ) -> ( ( x e. A -> ps ) <-> ( y e. B -> ch ) ) )
6 5 expcom 
 |-  ( x = y -> ( ph -> ( ( x e. A -> ps ) <-> ( y e. B -> ch ) ) ) )
7 6 pm5.74d 
 |-  ( x = y -> ( ( ph -> ( x e. A -> ps ) ) <-> ( ph -> ( y e. B -> ch ) ) ) )
8 7 cbvalvw 
 |-  ( A. x ( ph -> ( x e. A -> ps ) ) <-> A. y ( ph -> ( y e. B -> ch ) ) )
9 19.21v 
 |-  ( A. x ( ph -> ( x e. A -> ps ) ) <-> ( ph -> A. x ( x e. A -> ps ) ) )
10 19.21v 
 |-  ( A. y ( ph -> ( y e. B -> ch ) ) <-> ( ph -> A. y ( y e. B -> ch ) ) )
11 8 9 10 3bitr3i 
 |-  ( ( ph -> A. x ( x e. A -> ps ) ) <-> ( ph -> A. y ( y e. B -> ch ) ) )
12 11 pm5.74ri 
 |-  ( ph -> ( A. x ( x e. A -> ps ) <-> A. y ( y e. B -> ch ) ) )
13 df-ral 
 |-  ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
14 df-ral 
 |-  ( A. y e. B ch <-> A. y ( y e. B -> ch ) )
15 12 13 14 3bitr4g 
 |-  ( ph -> ( A. x e. A ps <-> A. y e. B ch ) )