Theorem caovcom 6163

Description: Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)

Hypotheses

Ref	Expression
caovcom.1	|- A e. _V
caovcom.2	|- B e. _V
caovcom.3	|- ( x F y ) = ( y F x )

Assertion

Ref	Expression
caovcom	|- ( A F B ) = ( B F A )

Distinct variable groups:   x, y, A   x, B, y   x, F, y

Proof of Theorem caovcom

Step	Hyp	Ref	Expression
1		caovcom.1	. 2 |- A e. _V
2		caovcom.2	. . 3 |- B e. _V
3	1, 2	pm3.2i 272	. 2 |- ( A e. _V /\ B e. _V )
4		caovcom.3	. . . 4 |- ( x F y ) = ( y F x )
5	4	a1i 9	. . 3 |- ( ( A e. _V /\ ( x e. _V /\ y e. _V ) ) -> ( x F y ) = ( y F x ) )
6	5	caovcomg 6161	. 2 |- ( ( A e. _V /\ ( A e. _V /\ B e. _V ) ) -> ( A F B ) = ( B F A ) )
7	1, 3, 6	mp2an 426	1 |- ( A F B ) = ( B F A )

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799  (class class class)co 6001

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004

This theorem is referenced by:  caovord2  6178  caov32  6193  caov12  6194  ecopovsym  6778  ecopover  6780