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Theorem caovassd 5930
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 5929 . 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 1218 1  |-  ( ph  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331    e. wcel 1480  (class class class)co 5774
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  caov32d  5951  caov12d  5952  caov13d  5954  caov4d  5955  caovdilemd  5962  caovimo  5964  grprinvlem  5965  grprinvd  5966  grpridd  5967  enq0tr  7249  prarloclemlo  7309  ltsosr  7579
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