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Metamath Proof Explorer

Theorem caovdi

Description: Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995) (Revised by Mario Carneiro, 28-Jun-2013)

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

Proof

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