Theorem caovord2 6178

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 6177	. 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 6163	. . 3 |- ( C F A ) = ( A F C )
8	5, 2, 6	caovcom 6163	. . 3 |- ( C F B ) = ( B F C )
9	7, 8	breq12i 4092	. 2 |- ( ( C F A ) R ( C F B ) <-> ( A F C ) R ( B F C ) )
10	4, 9	bitrdi 196	1 |- ( C e. S -> ( A R B <-> ( A F C ) R ( B F C ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   class class class wbr 4083  (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:  caovord3  6179