Theorem caovord2 6025

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 6024	. 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 6010	. . 3 |- ( C F A ) = ( A F C )
8	5, 2, 6	caovcom 6010	. . 3 |- ( C F B ) = ( B F C )
9	7, 8	breq12i 3998	. 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 1348   e. wcel 2141   _Vcvv 2730   class class class wbr 3989  (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:  caovord3  6026