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Theorem caovcomg 6242
Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)
Hypothesis
Ref Expression
caovcomg.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
Assertion
Ref Expression
caovcomg  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A F B )  =  ( B F A ) )
Distinct variable groups:    x, y, A    x, B, y    ph, x, y    x, F, y    x, S, y

Proof of Theorem caovcomg
StepHypRef Expression
1 caovcomg.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
21ralrimivva 2798 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x F y )  =  ( y F x ) )
3 oveq1 6088 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
4 oveq2 6089 . . . 4  |-  ( x  =  A  ->  (
y F x )  =  ( y F A ) )
53, 4eqeq12d 2450 . . 3  |-  ( x  =  A  ->  (
( x F y )  =  ( y F x )  <->  ( A F y )  =  ( y F A ) ) )
6 oveq2 6089 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
7 oveq1 6088 . . . 4  |-  ( y  =  B  ->  (
y F A )  =  ( B F A ) )
86, 7eqeq12d 2450 . . 3  |-  ( y  =  B  ->  (
( A F y )  =  ( y F A )  <->  ( A F B )  =  ( B F A ) ) )
95, 8rspc2v 3058 . 2  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A. x  e.  S  A. y  e.  S  ( x F y )  =  ( y F x )  ->  ( A F B )  =  ( B F A ) ) )
102, 9mpan9 456 1  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A F B )  =  ( B F A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705  (class class class)co 6081
This theorem is referenced by:  caovcomd  6243  caovcom  6244  caofcom  6336  seqcaopr  11360  cmncom  15428
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084
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