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Theorem caovdi 5708
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 1263 . . 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 5703 . . 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 408 . 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 1243 1  |-  ( A G ( B F C ) )  =  ( ( A G B ) F ( A G C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    /\ w3a 896    = wceq 1259   T. wtru 1260    e. wcel 1409   _Vcvv 2574  (class class class)co 5540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543
This theorem is referenced by: (None)
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