Theorem caovcomg 5816

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 2456	. 2 |- ( ph -> A. x e. S A. y e. S ( x F y ) = ( y F x ) )
3		oveq1 5675	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
4		oveq2 5676	. . . 4 |- ( x = A -> ( y F x ) = ( y F A ) )
5	3, 4	eqeq12d 2103	. . 3 |- ( x = A -> ( ( x F y ) = ( y F x ) <-> ( A F y ) = ( y F A ) ) )
6		oveq2 5676	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
7		oveq1 5675	. . . 4 |- ( y = B -> ( y F A ) = ( B F A ) )
8	6, 7	eqeq12d 2103	. . 3 |- ( y = B -> ( ( A F y ) = ( y F A ) <-> ( A F B ) = ( B F A ) ) )
9	5, 8	rspc2v 2737	. 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 276	1 |- ( ( ph /\ ( A e. S /\ B e. S ) ) -> ( A F B ) = ( B F A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   A.wral 2360  (class class class)co 5668

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-iota 4995  df-fv 5038  df-ov 5671

This theorem is referenced by:  caovcomd  5817  caovcom  5818  caovlem2d  5853  caofcom  5894  iseqcaopr  9975