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Theorem caovdi 5816
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 1293 . . 3  |- T.
5 caovdi.4 . . . . 5  |-  ( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) )
65a1i 9 . . . 4  |-  ( ( T.  /\  ( x  e.  _V  /\  y  e.  _V  /\  z  e. 
_V ) )  -> 
( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) ) )
76caovdig 5811 . . 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 415 . 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 1273 1  |-  ( A G ( B F C ) )  =  ( ( A G B ) F ( A G C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    /\ w3a 924    = wceq 1289   T. wtru 1290    e. wcel 1438   _Vcvv 2619  (class class class)co 5644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-iota 4975  df-fv 5018  df-ov 5647
This theorem is referenced by: (None)
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