Theorem caovord2d 6224

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 6223	. 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 6211	. . 3 |- ( ph -> ( C F A ) = ( A F C ) )
8	6, 4, 3	caovcomd 6211	. . 3 |- ( ph -> ( C F B ) = ( B F C ) )
9	7, 8	breq12d 4122	. 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 2203   class class class wbr 4109  (class class class)co 6050

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 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053

This theorem is referenced by:  caovord3d  6225  genplt2i  7825  addnqprllem  7842  addnqprulem  7843  mulnqprl  7883  mulnqpru  7884  distrlem4prl  7899  distrlem4pru  7900  1idprl  7905  1idpru  7906  ltexprlemdisj  7921  ltexprlemloc  7922  ltexprlemfl  7924  ltexprlemfu  7926  prplnqu  7935  recexprlem1ssl  7948  recexprlem1ssu  7949  aptiprleml  7954  aptiprlemu  7955  caucvgprlemcanl  7959  cauappcvgprlemlol  7962  cauappcvgprlemloc  7967  cauappcvgprlemladdfu  7969  cauappcvgprlemladdru  7971  cauappcvgprlemladdrl  7972  cauappcvgprlem1  7974  caucvgprlemnkj  7981  caucvgprlemnbj  7982  caucvgprlemlol  7985  caucvgprlemloc  7990  caucvgprlemladdfu  7992  caucvgprlemladdrl  7993  caucvgprprlemnkltj  8004  caucvgprprlemnbj  8008  caucvgprprlemmu  8010  caucvgprprlemlol  8013  caucvgprprlemloc  8018  caucvgprprlemexbt  8021  caucvgprprlemexb  8022  caucvgprprlemaddq  8023  lttrsr  8077  ltsosr  8079  prsrlt  8102  caucvgsrlemoffcau  8113  caucvgsrlemoffgt1  8114  caucvgsrlemoffres  8115  caucvgsr  8117