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Metamath Proof Explorer

Theorem ralxfrd

Description: Transfer universal quantification from a variable x to another variable y contained in expression A . (Contributed by NM, 15-Aug-2014) (Proof shortened by Mario Carneiro, 19-Nov-2016) (Proof shortened by JJ, 7-Aug-2021)

Ref Expression
Hypotheses ralxfrd.1 
|- ( ( ph /\ y e. C ) -> A e. B )
ralxfrd.2 
|- ( ( ph /\ x e. B ) -> E. y e. C x = A )
ralxfrd.3 
|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
Assertion ralxfrd 
|- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )

Proof

Step Hyp Ref Expression
1 ralxfrd.1 
 |-  ( ( ph /\ y e. C ) -> A e. B )
2 ralxfrd.2 
 |-  ( ( ph /\ x e. B ) -> E. y e. C x = A )
3 ralxfrd.3 
 |-  ( ( ph /\ x = A ) -> ( ps <-> ch ) )
4 3 adantlr 
 |-  ( ( ( ph /\ y e. C ) /\ x = A ) -> ( ps <-> ch ) )
5 1 4 rspcdv 
 |-  ( ( ph /\ y e. C ) -> ( A. x e. B ps -> ch ) )
6 5 ralrimdva 
 |-  ( ph -> ( A. x e. B ps -> A. y e. C ch ) )
7 r19.29 
 |-  ( ( A. y e. C ch /\ E. y e. C x = A ) -> E. y e. C ( ch /\ x = A ) )
8 3 exbiri 
 |-  ( ph -> ( x = A -> ( ch -> ps ) ) )
9 8 impcomd 
 |-  ( ph -> ( ( ch /\ x = A ) -> ps ) )
10 9 rexlimdvw 
 |-  ( ph -> ( E. y e. C ( ch /\ x = A ) -> ps ) )
11 7 10 syl5 
 |-  ( ph -> ( ( A. y e. C ch /\ E. y e. C x = A ) -> ps ) )
12 11 adantr 
 |-  ( ( ph /\ x e. B ) -> ( ( A. y e. C ch /\ E. y e. C x = A ) -> ps ) )
13 2 12 mpan2d 
 |-  ( ( ph /\ x e. B ) -> ( A. y e. C ch -> ps ) )
14 13 ralrimdva 
 |-  ( ph -> ( A. y e. C ch -> A. x e. B ps ) )
15 6 14 impbid 
 |-  ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )