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Metamath Proof Explorer

Theorem ceqsrex2v

Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005)

Ref Expression
Hypotheses ceqsrex2v.1 
|- ( x = A -> ( ph <-> ps ) )
ceqsrex2v.2 
|- ( y = B -> ( ps <-> ch ) )
Assertion ceqsrex2v 
|- ( ( A e. C /\ B e. D ) -> ( E. x e. C E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> ch ) )

Proof

Step Hyp Ref Expression
1 ceqsrex2v.1 
 |-  ( x = A -> ( ph <-> ps ) )
2 ceqsrex2v.2 
 |-  ( y = B -> ( ps <-> ch ) )
3 anass 
 |-  ( ( ( x = A /\ y = B ) /\ ph ) <-> ( x = A /\ ( y = B /\ ph ) ) )
4 3 rexbii 
 |-  ( E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> E. y e. D ( x = A /\ ( y = B /\ ph ) ) )
5 r19.42v 
 |-  ( E. y e. D ( x = A /\ ( y = B /\ ph ) ) <-> ( x = A /\ E. y e. D ( y = B /\ ph ) ) )
6 4 5 bitri 
 |-  ( E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> ( x = A /\ E. y e. D ( y = B /\ ph ) ) )
7 6 rexbii 
 |-  ( E. x e. C E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> E. x e. C ( x = A /\ E. y e. D ( y = B /\ ph ) ) )
8 1 anbi2d 
 |-  ( x = A -> ( ( y = B /\ ph ) <-> ( y = B /\ ps ) ) )
9 8 rexbidv 
 |-  ( x = A -> ( E. y e. D ( y = B /\ ph ) <-> E. y e. D ( y = B /\ ps ) ) )
10 9 ceqsrexv 
 |-  ( A e. C -> ( E. x e. C ( x = A /\ E. y e. D ( y = B /\ ph ) ) <-> E. y e. D ( y = B /\ ps ) ) )
11 7 10 bitrid 
 |-  ( A e. C -> ( E. x e. C E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> E. y e. D ( y = B /\ ps ) ) )
12 2 ceqsrexv 
 |-  ( B e. D -> ( E. y e. D ( y = B /\ ps ) <-> ch ) )
13 11 12 sylan9bb 
 |-  ( ( A e. C /\ B e. D ) -> ( E. x e. C E. y e. D ( ( x = A /\ y = B ) /\ ph ) <-> ch ) )