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Metamath Proof Explorer

Theorem ralxfr2d

Description: Transfer universal quantification from a variable x to another variable y contained in expression A . (Contributed by Mario Carneiro, 20-Aug-2014)

Ref Expression
Hypotheses ralxfr2d.1 
|- ( ( ph /\ y e. C ) -> A e. V )
ralxfr2d.2 
|- ( ph -> ( x e. B <-> E. y e. C x = A ) )
ralxfr2d.3 
|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
Assertion ralxfr2d 
|- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )

Proof

Step Hyp Ref Expression
1 ralxfr2d.1 
 |-  ( ( ph /\ y e. C ) -> A e. V )
2 ralxfr2d.2 
 |-  ( ph -> ( x e. B <-> E. y e. C x = A ) )
3 ralxfr2d.3 
 |-  ( ( ph /\ x = A ) -> ( ps <-> ch ) )
4 elisset 
 |-  ( A e. V -> E. x x = A )
5 1 4 syl 
 |-  ( ( ph /\ y e. C ) -> E. x x = A )
6 2 biimprd 
 |-  ( ph -> ( E. y e. C x = A -> x e. B ) )
7 r19.23v 
 |-  ( A. y e. C ( x = A -> x e. B ) <-> ( E. y e. C x = A -> x e. B ) )
8 6 7 sylibr 
 |-  ( ph -> A. y e. C ( x = A -> x e. B ) )
9 8 r19.21bi 
 |-  ( ( ph /\ y e. C ) -> ( x = A -> x e. B ) )
10 eleq1 
 |-  ( x = A -> ( x e. B <-> A e. B ) )
11 9 10 mpbidi 
 |-  ( ( ph /\ y e. C ) -> ( x = A -> A e. B ) )
12 11 exlimdv 
 |-  ( ( ph /\ y e. C ) -> ( E. x x = A -> A e. B ) )
13 5 12 mpd 
 |-  ( ( ph /\ y e. C ) -> A e. B )
14 2 biimpa 
 |-  ( ( ph /\ x e. B ) -> E. y e. C x = A )
15 13 14 3 ralxfrd 
 |-  ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )