Theorem caovcomd 6103

Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovcomg.1	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
caovcomd.2	|- ( ph -> A e. S )
caovcomd.3	|- ( ph -> B e. S )

Assertion

Ref	Expression
caovcomd	|- ( ph -> ( 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 caovcomd

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( ph -> ph )
2		caovcomd.2	. 2 |- ( ph -> A e. S )
3		caovcomd.3	. 2 |- ( ph -> B e. S )
4		caovcomg.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
5	4	caovcomg 6102	. 2 |- ( ( ph /\ ( A e. S /\ B e. S ) ) -> ( A F B ) = ( B F A ) )
6	1, 2, 3, 5	syl12anc 1248	1 |- ( ph -> ( A F B ) = ( B F A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176  (class class class)co 5944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  caovcanrd  6110  caovord2d  6116  caovdir2d  6123  caov32d  6127  caov12d  6128  caov31d  6129  caov411d  6132  caov42d  6133  caovimo  6140  ecopovsymg  6721  ecopoverg  6723  ltsonq  7511  prarloclemlo  7607  addextpr  7734  ltsosr  7877  ltasrg  7883  mulextsr1lem  7893  seq3f1olemqsumkj  10656  seqf1oglem2a  10663