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Theorem caovdir 6402
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 6387 . 2  |-  ( C G ( A F B ) )  =  ( ( C G A ) F ( C G B ) )
6 ovex 6220 . . 3  |-  ( A F B )  e. 
_V
7 caovdir.com . . 3  |-  ( x G y )  =  ( y G x )
81, 6, 7caovcom 6365 . 2  |-  ( C G ( A F B ) )  =  ( ( A F B ) G C )
91, 2, 7caovcom 6365 . . 3  |-  ( C G A )  =  ( A G C )
101, 3, 7caovcom 6365 . . 3  |-  ( C G B )  =  ( B G C )
119, 10oveq12i 6207 . 2  |-  ( ( C G A ) F ( C G B ) )  =  ( ( A G C ) F ( B G C ) )
125, 8, 113eqtr3i 2489 1  |-  ( ( A F B ) G C )  =  ( ( A G C ) F ( B G C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   _Vcvv 3072  (class class class)co 6195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-nul 4524
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-iota 5484  df-fv 5529  df-ov 6198
This theorem is referenced by:  caovdilem  6403  adderpqlem  9229  addassnq  9233  prlem934  9308  prlem936  9322  recexsrlem  9376  mulgt0sr  9378
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