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Metamath Proof Explorer

Theorem caovcan

Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995)

Ref Expression
Hypotheses caovcan.1 
|- C e. _V
caovcan.2 
|- ( ( x e. S /\ y e. S ) -> ( ( x F y ) = ( x F z ) -> y = z ) )
Assertion caovcan 
|- ( ( A e. S /\ B e. S ) -> ( ( A F B ) = ( A F C ) -> B = C ) )

Proof

Step Hyp Ref Expression
1 caovcan.1 
 |-  C e. _V
2 caovcan.2 
 |-  ( ( x e. S /\ y e. S ) -> ( ( x F y ) = ( x F z ) -> y = z ) )
3 oveq1 
 |-  ( x = A -> ( x F y ) = ( A F y ) )
4 oveq1 
 |-  ( x = A -> ( x F C ) = ( A F C ) )
5 3 4 eqeq12d 
 |-  ( x = A -> ( ( x F y ) = ( x F C ) <-> ( A F y ) = ( A F C ) ) )
6 5 imbi1d 
 |-  ( x = A -> ( ( ( x F y ) = ( x F C ) -> y = C ) <-> ( ( A F y ) = ( A F C ) -> y = C ) ) )
7 oveq2 
 |-  ( y = B -> ( A F y ) = ( A F B ) )
8 7 eqeq1d 
 |-  ( y = B -> ( ( A F y ) = ( A F C ) <-> ( A F B ) = ( A F C ) ) )
9 eqeq1 
 |-  ( y = B -> ( y = C <-> B = C ) )
10 8 9 imbi12d 
 |-  ( y = B -> ( ( ( A F y ) = ( A F C ) -> y = C ) <-> ( ( A F B ) = ( A F C ) -> B = C ) ) )
11 oveq2 
 |-  ( z = C -> ( x F z ) = ( x F C ) )
12 11 eqeq2d 
 |-  ( z = C -> ( ( x F y ) = ( x F z ) <-> ( x F y ) = ( x F C ) ) )
13 eqeq2 
 |-  ( z = C -> ( y = z <-> y = C ) )
14 12 13 imbi12d 
 |-  ( z = C -> ( ( ( x F y ) = ( x F z ) -> y = z ) <-> ( ( x F y ) = ( x F C ) -> y = C ) ) )
15 14 imbi2d 
 |-  ( z = C -> ( ( ( x e. S /\ y e. S ) -> ( ( x F y ) = ( x F z ) -> y = z ) ) <-> ( ( x e. S /\ y e. S ) -> ( ( x F y ) = ( x F C ) -> y = C ) ) ) )
16 1 15 2 vtocl 
 |-  ( ( x e. S /\ y e. S ) -> ( ( x F y ) = ( x F C ) -> y = C ) )
17 6 10 16 vtocl2ga 
 |-  ( ( A e. S /\ B e. S ) -> ( ( A F B ) = ( A F C ) -> B = C ) )