Theorem caovord2d 6232

Description: Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovordg.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )
caovordd.2	|- ( ph -> A e. S )
caovordd.3	|- ( ph -> B e. S )
caovordd.4	|- ( ph -> C e. S )
caovord2d.com	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )

Assertion

Ref	Expression
caovord2d	|- ( ph -> ( 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   ph, x, y, z   x, F, y, z   x, R, y, z   x, S, y, z

Proof of Theorem caovord2d

Step	Hyp	Ref	Expression
1		caovordg.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )
2		caovordd.2	. . 3 |- ( ph -> A e. S )
3		caovordd.3	. . 3 |- ( ph -> B e. S )
4		caovordd.4	. . 3 |- ( ph -> C e. S )
5	1, 2, 3, 4	caovordd 6231	. 2 |- ( ph -> ( A R B <-> ( C F A ) R ( C F B ) ) )
6		caovord2d.com	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
7	6, 4, 2	caovcomd 6219	. . 3 |- ( ph -> ( C F A ) = ( A F C ) )
8	6, 4, 3	caovcomd 6219	. . 3 |- ( ph -> ( C F B ) = ( B F C ) )
9	7, 8	breq12d 4127	. 2 |- ( ph -> ( ( C F A ) R ( C F B ) <-> ( A F C ) R ( B F C ) ) )
10	5, 9	bitrd 188	1 |- ( ph -> ( A R B <-> ( A F C ) R ( B F C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   class class class wbr 4114  (class class class)co 6058

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061

This theorem is referenced by:  caovord3d  6233  genplt2i  7841  addnqprllem  7858  addnqprulem  7859  mulnqprl  7899  mulnqpru  7900  distrlem4prl  7915  distrlem4pru  7916  1idprl  7921  1idpru  7922  ltexprlemdisj  7937  ltexprlemloc  7938  ltexprlemfl  7940  ltexprlemfu  7942  prplnqu  7951  recexprlem1ssl  7964  recexprlem1ssu  7965  aptiprleml  7970  aptiprlemu  7971  caucvgprlemcanl  7975  cauappcvgprlemlol  7978  cauappcvgprlemloc  7983  cauappcvgprlemladdfu  7985  cauappcvgprlemladdru  7987  cauappcvgprlemladdrl  7988  cauappcvgprlem1  7990  caucvgprlemnkj  7997  caucvgprlemnbj  7998  caucvgprlemlol  8001  caucvgprlemloc  8006  caucvgprlemladdfu  8008  caucvgprlemladdrl  8009  caucvgprprlemnkltj  8020  caucvgprprlemnbj  8024  caucvgprprlemmu  8026  caucvgprprlemlol  8029  caucvgprprlemloc  8034  caucvgprprlemexbt  8037  caucvgprprlemexb  8038  caucvgprprlemaddq  8039  lttrsr  8093  ltsosr  8095  prsrlt  8118  caucvgsrlemoffcau  8129  caucvgsrlemoffgt1  8130  caucvgsrlemoffres  8131  caucvgsr  8133