Theorem caovord2d 6223

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 6222	. 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 6210	. . 3 |- ( ph -> ( C F A ) = ( A F C ) )
8	6, 4, 3	caovcomd 6210	. . 3 |- ( ph -> ( C F B ) = ( B F C ) )
9	7, 8	breq12d 4121	. 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 4108  (class class class)co 6049

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 2814  df-un 3214  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-iota 5311  df-fv 5359  df-ov 6052

This theorem is referenced by:  caovord3d  6224  genplt2i  7824  addnqprllem  7841  addnqprulem  7842  mulnqprl  7882  mulnqpru  7883  distrlem4prl  7898  distrlem4pru  7899  1idprl  7904  1idpru  7905  ltexprlemdisj  7920  ltexprlemloc  7921  ltexprlemfl  7923  ltexprlemfu  7925  prplnqu  7934  recexprlem1ssl  7947  recexprlem1ssu  7948  aptiprleml  7953  aptiprlemu  7954  caucvgprlemcanl  7958  cauappcvgprlemlol  7961  cauappcvgprlemloc  7966  cauappcvgprlemladdfu  7968  cauappcvgprlemladdru  7970  cauappcvgprlemladdrl  7971  cauappcvgprlem1  7973  caucvgprlemnkj  7980  caucvgprlemnbj  7981  caucvgprlemlol  7984  caucvgprlemloc  7989  caucvgprlemladdfu  7991  caucvgprlemladdrl  7992  caucvgprprlemnkltj  8003  caucvgprprlemnbj  8007  caucvgprprlemmu  8009  caucvgprprlemlol  8012  caucvgprprlemloc  8017  caucvgprprlemexbt  8020  caucvgprprlemexb  8021  caucvgprprlemaddq  8022  lttrsr  8076  ltsosr  8078  prsrlt  8101  caucvgsrlemoffcau  8112  caucvgsrlemoffgt1  8113  caucvgsrlemoffres  8114  caucvgsr  8116