Theorem caovcom 6010

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 6008	. 2 |- ( ( A e. _V /\ ( A e. _V /\ B e. _V ) ) -> ( A F B ) = ( B F A ) )
7	1, 3, 6	mp2an 424	1 |- ( A F B ) = ( B F A )

Syntax hints:   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730  (class class class)co 5853

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  caovord2  6025  caov32  6040  caov12  6041  ecopovsym  6609  ecopover  6611