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Metamath Proof Explorer

Theorem rexxfrd2

Description: Transfer existence from a variable x to another variable y contained in expression A . Variant of rexxfrd . (Contributed by Alexander van der Vekens, 25-Apr-2018)

Ref Expression
Hypotheses ralxfrd2.1 
|- ( ( ph /\ y e. C ) -> A e. B )
ralxfrd2.2 
|- ( ( ph /\ x e. B ) -> E. y e. C x = A )
ralxfrd2.3 
|- ( ( ph /\ y e. C /\ x = A ) -> ( ps <-> ch ) )
Assertion rexxfrd2 
|- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) )

Proof

Step Hyp Ref Expression
1 ralxfrd2.1 
 |-  ( ( ph /\ y e. C ) -> A e. B )
2 ralxfrd2.2 
 |-  ( ( ph /\ x e. B ) -> E. y e. C x = A )
3 ralxfrd2.3 
 |-  ( ( ph /\ y e. C /\ x = A ) -> ( ps <-> ch ) )
4 3 notbid 
 |-  ( ( ph /\ y e. C /\ x = A ) -> ( -. ps <-> -. ch ) )
5 1 2 4 ralxfrd2 
 |-  ( ph -> ( A. x e. B -. ps <-> A. y e. C -. ch ) )
6 5 notbid 
 |-  ( ph -> ( -. A. x e. B -. ps <-> -. A. y e. C -. ch ) )
7 dfrex2 
 |-  ( E. x e. B ps <-> -. A. x e. B -. ps )
8 dfrex2 
 |-  ( E. y e. C ch <-> -. A. y e. C -. ch )
9 6 7 8 3bitr4g 
 |-  ( ph -> ( E. x e. B ps <-> E. y e. C ch ) )