Theorem caovcomg 6052

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

Step	Hyp	Ref	Expression
1		caovcomg.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
2	1	ralrimivva 2572	. 2 |- ( ph -> A. x e. S A. y e. S ( x F y ) = ( y F x ) )
3		oveq1 5903	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
4		oveq2 5904	. . . 4 |- ( x = A -> ( y F x ) = ( y F A ) )
5	3, 4	eqeq12d 2204	. . 3 |- ( x = A -> ( ( x F y ) = ( y F x ) <-> ( A F y ) = ( y F A ) ) )
6		oveq2 5904	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
7		oveq1 5903	. . . 4 |- ( y = B -> ( y F A ) = ( B F A ) )
8	6, 7	eqeq12d 2204	. . 3 |- ( y = B -> ( ( A F y ) = ( y F A ) <-> ( A F B ) = ( B F A ) ) )
9	5, 8	rspc2v 2869	. 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 ) ) )
10	2, 9	mpan9 281	1 |- ( ( ph /\ ( A e. S /\ B e. S ) ) -> ( A F B ) = ( B F A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   A.wral 2468  (class class class)co 5896

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5899

This theorem is referenced by:  caovcomd  6053  caovcom  6054  caovlem2d  6089  caofcom  6130  seq3caopr  10514  cmncom  13241