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Theorem caovassd 6395
Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovassg.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
caovassd.2  |-  ( ph  ->  A  e.  S )
caovassd.3  |-  ( ph  ->  B  e.  S )
caovassd.4  |-  ( ph  ->  C  e.  S )
Assertion
Ref Expression
caovassd  |-  ( ph  ->  ( ( A F B ) F C )  =  ( A F ( B F 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, S, y, z

Proof of Theorem caovassd
StepHypRef Expression
1 id 22 . 2  |-  ( ph  ->  ph )
2 caovassd.2 . 2  |-  ( ph  ->  A  e.  S )
3 caovassd.3 . 2  |-  ( ph  ->  B  e.  S )
4 caovassd.4 . 2  |-  ( ph  ->  C  e.  S )
5 caovassg.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
65caovassg 6394 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) )  -> 
( ( A F B ) F C )  =  ( A F ( B F C ) ) )
71, 2, 3, 4, 6syl13anc 1228 1  |-  ( ph  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836  (class class class)co 6218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-iota 5477  df-fv 5521  df-ov 6221
This theorem is referenced by:  caov32d  6416  caov12d  6417  caov13d  6419  caov4d  6420  grprinvlem  6434  grprinvd  6435  grpridd  6436  seqf1olem2a  12071  grprcan  16223
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