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Theorem caovdid 5953
Description: Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovdig.1  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) ) )
caovdid.2  |-  ( ph  ->  A  e.  K )
caovdid.3  |-  ( ph  ->  B  e.  S )
caovdid.4  |-  ( ph  ->  C  e.  S )
Assertion
Ref Expression
caovdid  |-  ( ph  ->  ( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) )
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    ph, x, y, z   
x, F, y, z   
x, G, y, z   
x, H, y, z   
x, K, y, z   
x, S, y, z

Proof of Theorem caovdid
StepHypRef Expression
1 id 19 . 2  |-  ( ph  ->  ph )
2 caovdid.2 . 2  |-  ( ph  ->  A  e.  K )
3 caovdid.3 . 2  |-  ( ph  ->  B  e.  S )
4 caovdid.4 . 2  |-  ( ph  ->  C  e.  S )
5 caovdig.1 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) ) )
65caovdig 5952 . 2  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  S  /\  C  e.  S ) )  -> 
( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) )
71, 2, 3, 4, 6syl13anc 1219 1  |-  ( ph  ->  ( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    = wceq 1332    e. wcel 1481  (class class class)co 5781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-iota 5095  df-fv 5138  df-ov 5784
This theorem is referenced by:  caovdir2d  5954  caovlem2d  5970  ltanqg  7231
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