Theorem caovcom 5658

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 257	. 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 5656	. 2 |- ( ( A e. _V /\ ( A e. _V /\ B e. _V ) ) -> ( A F B ) = ( B F A ) )
7	1, 3, 6	mp2an 402	1 |- ( A F B ) = ( B F A )

Syntax hints:   /\ wa 97   = wceq 1243   e. wcel 1393   _Vcvv 2557  (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:  caovord2  5673  caov32  5688  caov12  5689  ecopovsym  6202  ecopover  6204