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Theorem caovdi 6039
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 1312 . . 3  |-  T.
5 caovdi.4 . . . . 5  |-  ( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) )
65a1i 10 . . . 4  |-  ( (  T.  /\  ( x  e.  _V  /\  y  e.  _V  /\  z  e. 
_V ) )  -> 
( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) ) )
76caovdig 6034 . . 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 651 . 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 1277 1  |-  ( A G ( B F C ) )  =  ( ( A G B ) F ( A G C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    /\ w3a 934    T. wtru 1307    = wceq 1623    e. wcel 1684   _Vcvv 2788  (class class class)co 5858
This theorem is referenced by:  caovdir  6054  caovlem2  6056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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