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Theorem caovdid 6254
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 20 . 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 6253 . 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 1186 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725  (class class class)co 6073
This theorem is referenced by:  caovdir2d  6255  caofdi  6332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076
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