Theorem caovcomg 5656

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 2401	. 2 |- ( ph -> A. x e. S A. y e. S ( x F y ) = ( y F x ) )
3		oveq1 5519	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
4		oveq2 5520	. . . 4 |- ( x = A -> ( y F x ) = ( y F A ) )
5	3, 4	eqeq12d 2054	. . 3 |- ( x = A -> ( ( x F y ) = ( y F x ) <-> ( A F y ) = ( y F A ) ) )
6		oveq2 5520	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
7		oveq1 5519	. . . 4 |- ( y = B -> ( y F A ) = ( B F A ) )
8	6, 7	eqeq12d 2054	. . 3 |- ( y = B -> ( ( A F y ) = ( y F A ) <-> ( A F B ) = ( B F A ) ) )
9	5, 8	rspc2v 2662	. 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 265	1 |- ( ( ph /\ ( A e. S /\ B e. S ) ) -> ( A F B ) = ( B F A ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   A.wral 2306  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  caovcomd  5657  caovcom  5658  caovlem2d  5693  caofcom  5734  iseqcaopr  9242