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Theorem caovassd 6146
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 19 . 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 6145 . 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 1185 1  |-  ( ph  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715  (class class class)co 5981
This theorem is referenced by:  caov32d  6167  caov12d  6168  caov13d  6170  caov4d  6171  grprinvlem  6185  grprinvd  6186  grpridd  6187  seqf1olem2a  11248  grprcan  14725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-iota 5322  df-fv 5366  df-ov 5984
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