Theorem caovdir 6868

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

Hypotheses

Ref	Expression
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

Ref	Expression
caovdir	|- ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   x, F, y, z   x, G, y, z

Proof of Theorem caovdir

Step	Hyp	Ref	Expression
1		caovdir.3	. . 3 |- C e. _V
2		caovdir.1	. . 3 |- A e. _V
3		caovdir.2	. . 3 |- B e. _V
4		caovdir.distr	. . 3 |- ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) )
5	1, 2, 3, 4	caovdi 6853	. 2 |- ( C G ( A F B ) ) = ( ( C G A ) F ( C G B ) )
6		ovex 6678	. . 3 |- ( A F B ) e. _V
7		caovdir.com	. . 3 |- ( x G y ) = ( y G x )
8	1, 6, 7	caovcom 6831	. 2 |- ( C G ( A F B ) ) = ( ( A F B ) G C )
9	1, 2, 7	caovcom 6831	. . 3 |- ( C G A ) = ( A G C )
10	1, 3, 7	caovcom 6831	. . 3 |- ( C G B ) = ( B G C )
11	9, 10	oveq12i 6662	. 2 |- ( ( C G A ) F ( C G B ) ) = ( ( A G C ) F ( B G C ) )
12	5, 8, 11	3eqtr3i 2652	1 |- ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) )

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  caovdilem  6869  adderpqlem  9776  addassnq  9780  prlem934  9855  prlem936  9869  recexsrlem  9924  mulgt0sr  9926