Theorem caovdir

Description: Reverse distributive law. (Contributed by NM, 26-Aug-1995)

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

Proof

Step	Hyp	Ref	Expression
1		caovdir.1	|- A e. _V
2		caovdir.2	|- B e. _V
3		caovdir.3	|- C e. _V
4		caovdir.com	|- ( x G y ) = ( y G x )
5		caovdir.distr	|- ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) )
6	3 1 2 5	caovdi	|- ( C G ( A F B ) ) = ( ( C G A ) F ( C G B ) )
7		ovex	|- ( A F B ) e. _V
8	3 7 4	caovcom	|- ( C G ( A F B ) ) = ( ( A F B ) G C )
9	3 1 4	caovcom	|- ( C G A ) = ( A G C )
10	3 2 4	caovcom	|- ( C G B ) = ( B G C )
11	9 10	oveq12i	|- ( ( C G A ) F ( C G B ) ) = ( ( A G C ) F ( B G C ) )
12	6 8 11	3eqtr3i	|- ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) )