Theorem caovord2 6014

Description: Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)

Hypotheses

Ref	Expression
caovord.1	|- A e. _V
caovord.2	|- B e. _V
caovord.3	|- ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) )
caovord2.3	|- C e. _V
caovord2.com	|- ( x F y ) = ( y F x )

Assertion

Ref	Expression
caovord2	|- ( C e. S -> ( A R B <-> ( A F C ) R ( B F C ) ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   x, F, y, z   x, R, y, z   x, S, y, z

Proof of Theorem caovord2

Step	Hyp	Ref	Expression
1		caovord.1	. . 3 |- A e. _V
2		caovord.2	. . 3 |- B e. _V
3		caovord.3	. . 3 |- ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) )
4	1, 2, 3	caovord 6013	. 2 |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )
5		caovord2.3	. . . 4 |- C e. _V
6		caovord2.com	. . . 4 |- ( x F y ) = ( y F x )
7	5, 1, 6	caovcom 5999	. . 3 |- ( C F A ) = ( A F C )
8	5, 2, 6	caovcom 5999	. . 3 |- ( C F B ) = ( B F C )
9	7, 8	breq12i 3991	. 2 |- ( ( C F A ) R ( C F B ) <-> ( A F C ) R ( B F C ) )
10	4, 9	bitrdi 195	1 |- ( C e. S -> ( A R B <-> ( A F C ) R ( B F C ) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   class class class wbr 3982  (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:  caovord3  6015