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Metamath Proof Explorer

Theorem caovassg

Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013) (Revised by Mario Carneiro, 26-May-2014)

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

Proof

Step Hyp Ref Expression
1 caovassg.1 
 |-  ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
2 1 ralrimivvva 
 |-  ( ph -> A. x e. S A. y e. S A. z e. S ( ( x F y ) F z ) = ( x F ( y F z ) ) )
3 oveq1 
 |-  ( x = A -> ( x F y ) = ( A F y ) )
4 3 oveq1d 
 |-  ( x = A -> ( ( x F y ) F z ) = ( ( A F y ) F z ) )
5 oveq1 
 |-  ( x = A -> ( x F ( y F z ) ) = ( A F ( y F z ) ) )
6 4 5 eqeq12d 
 |-  ( x = A -> ( ( ( x F y ) F z ) = ( x F ( y F z ) ) <-> ( ( A F y ) F z ) = ( A F ( y F z ) ) ) )
7 oveq2 
 |-  ( y = B -> ( A F y ) = ( A F B ) )
8 7 oveq1d 
 |-  ( y = B -> ( ( A F y ) F z ) = ( ( A F B ) F z ) )
9 oveq1 
 |-  ( y = B -> ( y F z ) = ( B F z ) )
10 9 oveq2d 
 |-  ( y = B -> ( A F ( y F z ) ) = ( A F ( B F z ) ) )
11 8 10 eqeq12d 
 |-  ( y = B -> ( ( ( A F y ) F z ) = ( A F ( y F z ) ) <-> ( ( A F B ) F z ) = ( A F ( B F z ) ) ) )
12 oveq2 
 |-  ( z = C -> ( ( A F B ) F z ) = ( ( A F B ) F C ) )
13 oveq2 
 |-  ( z = C -> ( B F z ) = ( B F C ) )
14 13 oveq2d 
 |-  ( z = C -> ( A F ( B F z ) ) = ( A F ( B F C ) ) )
15 12 14 eqeq12d 
 |-  ( z = C -> ( ( ( A F B ) F z ) = ( A F ( B F z ) ) <-> ( ( A F B ) F C ) = ( A F ( B F C ) ) ) )
16 6 11 15 rspc3v 
 |-  ( ( A e. S /\ B e. S /\ C e. S ) -> ( A. x e. S A. y e. S A. z e. S ( ( x F y ) F z ) = ( x F ( y F z ) ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) ) )
17 2 16 mpan9 
 |-  ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )