Theorem caovcomg 6032

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 2559	. 2 |- ( ph -> A. x e. S A. y e. S ( x F y ) = ( y F x ) )
3		oveq1 5884	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
4		oveq2 5885	. . . 4 |- ( x = A -> ( y F x ) = ( y F A ) )
5	3, 4	eqeq12d 2192	. . 3 |- ( x = A -> ( ( x F y ) = ( y F x ) <-> ( A F y ) = ( y F A ) ) )
6		oveq2 5885	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
7		oveq1 5884	. . . 4 |- ( y = B -> ( y F A ) = ( B F A ) )
8	6, 7	eqeq12d 2192	. . 3 |- ( y = B -> ( ( A F y ) = ( y F A ) <-> ( A F B ) = ( B F A ) ) )
9	5, 8	rspc2v 2856	. 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 1353   e. wcel 2148   A.wral 2455  (class class class)co 5877

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880

This theorem is referenced by:  caovcomd  6033  caovcom  6034  caovlem2d  6069  caofcom  6108  seq3caopr  10485  cmncom  13110