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Theorem caovdi 6467
Description: Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.)
Hypotheses
Ref Expression
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
Ref Expression
caovdi  |-  ( A G ( B F C ) )  =  ( ( A G B ) F ( A 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 caovdi
StepHypRef Expression
1 caovdi.1 . 2  |-  A  e. 
_V
2 caovdi.2 . 2  |-  B  e. 
_V
3 caovdi.3 . 2  |-  C  e. 
_V
4 tru 1402 . . 3  |- T.
5 caovdi.4 . . . . 5  |-  ( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) )
65a1i 11 . . . 4  |-  ( ( T.  /\  ( x  e.  _V  /\  y  e.  _V  /\  z  e. 
_V ) )  -> 
( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) ) )
76caovdig 6462 . . 3  |-  ( ( T.  /\  ( A  e.  _V  /\  B  e.  _V  /\  C  e. 
_V ) )  -> 
( A G ( B F C ) )  =  ( ( A G B ) F ( A G C ) ) )
84, 7mpan 668 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( A G ( B F C ) )  =  ( ( A G B ) F ( A G C ) ) )
91, 2, 3, 8mp3an 1322 1  |-  ( A G ( B F C ) )  =  ( ( A G B ) F ( A G C ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    /\ w3a 971    = wceq 1398   T. wtru 1399    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
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-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  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:  caovdir  6482  caovlem2  6484
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