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Theorem caovdir 6482
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
StepHypRef 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 ) )
51, 2, 3, 4caovdi 6467 . 2  |-  ( C G ( A F B ) )  =  ( ( C G A ) F ( C G B ) )
6 ovex 6298 . . 3  |-  ( A F B )  e. 
_V
7 caovdir.com . . 3  |-  ( x G y )  =  ( y G x )
81, 6, 7caovcom 6445 . 2  |-  ( C G ( A F B ) )  =  ( ( A F B ) G C )
91, 2, 7caovcom 6445 . . 3  |-  ( C G A )  =  ( A G C )
101, 3, 7caovcom 6445 . . 3  |-  ( C G B )  =  ( B G C )
119, 10oveq12i 6282 . 2  |-  ( ( C G A ) F ( C G B ) )  =  ( ( A G C ) F ( B G C ) )
125, 8, 113eqtr3i 2491 1  |-  ( ( A F B ) G C )  =  ( ( A G C ) F ( B G C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823   _Vcvv 3106  (class class class)co 6270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  caovdilem  6483  adderpqlem  9321  addassnq  9325  prlem934  9400  prlem936  9414  recexsrlem  9469  mulgt0sr  9471
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