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Metamath Proof Explorer

Theorem caovass

Description: Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995) (Revised by Mario Carneiro, 26-May-2014)

Ref Expression
Hypotheses caovass.1 
|- A e. _V
caovass.2 
|- B e. _V
caovass.3 
|- C e. _V
caovass.4 
|- ( ( x F y ) F z ) = ( x F ( y F z ) )
Assertion caovass 
|- ( ( A F B ) F C ) = ( A F ( B F C ) )

Proof

Step Hyp Ref Expression
1 caovass.1 
 |-  A e. _V
2 caovass.2 
 |-  B e. _V
3 caovass.3 
 |-  C e. _V
4 caovass.4 
 |-  ( ( x F y ) F z ) = ( x F ( y F z ) )
5 tru 
 |-  T.
6 4 a1i 
 |-  ( ( T. /\ ( x e. _V /\ y e. _V /\ z e. _V ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
7 6 caovassg 
 |-  ( ( T. /\ ( A e. _V /\ B e. _V /\ C e. _V ) ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
8 5 7 mpan 
 |-  ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
9 1 2 3 8 mp3an 
 |-  ( ( A F B ) F C ) = ( A F ( B F C ) )