This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem caovcang

Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995) (Revised by Mario Carneiro, 30-Dec-2014)

Ref Expression
Hypothesis caovcang.1 
|- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x F y ) = ( x F z ) <-> y = z ) )
Assertion caovcang 
|- ( ( ph /\ ( A e. T /\ B e. S /\ C e. S ) ) -> ( ( A F B ) = ( A F C ) <-> B = C ) )

Proof

Step Hyp Ref Expression
1 caovcang.1 
 |-  ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x F y ) = ( x F z ) <-> y = z ) )
2 1 ralrimivvva 
 |-  ( ph -> A. x e. T A. y e. S A. z 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 z ) = ( A F z ) )
5 3 4 eqeq12d 
 |-  ( x = A -> ( ( x F y ) = ( x F z ) <-> ( A F y ) = ( A F z ) ) )
6 5 bibi1d 
 |-  ( x = A -> ( ( ( x F y ) = ( x F z ) <-> y = z ) <-> ( ( A F y ) = ( A F z ) <-> y = z ) ) )
7 oveq2 
 |-  ( y = B -> ( A F y ) = ( A F B ) )
8 7 eqeq1d 
 |-  ( y = B -> ( ( A F y ) = ( A F z ) <-> ( A F B ) = ( A F z ) ) )
9 eqeq1 
 |-  ( y = B -> ( y = z <-> B = z ) )
10 8 9 bibi12d 
 |-  ( y = B -> ( ( ( A F y ) = ( A F z ) <-> y = z ) <-> ( ( A F B ) = ( A F z ) <-> B = z ) ) )
11 oveq2 
 |-  ( z = C -> ( A F z ) = ( A F C ) )
12 11 eqeq2d 
 |-  ( z = C -> ( ( A F B ) = ( A F z ) <-> ( A F B ) = ( A F C ) ) )
13 eqeq2 
 |-  ( z = C -> ( B = z <-> B = C ) )
14 12 13 bibi12d 
 |-  ( z = C -> ( ( ( A F B ) = ( A F z ) <-> B = z ) <-> ( ( A F B ) = ( A F C ) <-> B = C ) ) )
15 6 10 14 rspc3v 
 |-  ( ( A e. T /\ B e. S /\ C e. S ) -> ( A. x e. T A. y e. S A. z e. S ( ( x F y ) = ( x F z ) <-> y = z ) -> ( ( A F B ) = ( A F C ) <-> B = C ) ) )
16 2 15 mpan9 
 |-  ( ( ph /\ ( A e. T /\ B e. S /\ C e. S ) ) -> ( ( A F B ) = ( A F C ) <-> B = C ) )