Theorem caovcom 5999

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 270	. 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 5997	. 2 |- ( ( A e. _V /\ ( A e. _V /\ B e. _V ) ) -> ( A F B ) = ( B F A ) )
7	1, 3, 6	mp2an 423	1 |- ( A F B ) = ( B F A )

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726  (class class class)co 5842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  caovord2  6014  caov32  6029  caov12  6030  ecopovsym  6597  ecopover  6599