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Theorem caovassd 6213
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 20 . 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 6212 . 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 1186 1  |-  ( ph  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721  (class class class)co 6048
This theorem is referenced by:  caov32d  6234  caov12d  6235  caov13d  6237  caov4d  6238  grprinvlem  6252  grprinvd  6253  grpridd  6254  seqf1olem2a  11324  grprcan  14801
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-ov 6051
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