Theorem caovord2d 5814

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 5813	. 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 5801	. . 3 |- ( ph -> ( C F A ) = ( A F C ) )
8	6, 4, 3	caovcomd 5801	. . 3 |- ( ph -> ( C F B ) = ( B F C ) )
9	7, 8	breq12d 3858	. 2 |- ( ph -> ( ( C F A ) R ( C F B ) <-> ( A F C ) R ( B F C ) ) )
10	5, 9	bitrd 186	1 |- ( ph -> ( A R B <-> ( A F C ) R ( B F C ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 924   = wceq 1289   e. wcel 1438   class class class wbr 3845  (class class class)co 5652

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655

This theorem is referenced by:  caovord3d  5815  genplt2i  7069  addnqprllem  7086  addnqprulem  7087  mulnqprl  7127  mulnqpru  7128  distrlem4prl  7143  distrlem4pru  7144  1idprl  7149  1idpru  7150  ltexprlemdisj  7165  ltexprlemloc  7166  ltexprlemfl  7168  ltexprlemfu  7170  prplnqu  7179  recexprlem1ssl  7192  recexprlem1ssu  7193  aptiprleml  7198  aptiprlemu  7199  caucvgprlemcanl  7203  cauappcvgprlemlol  7206  cauappcvgprlemloc  7211  cauappcvgprlemladdfu  7213  cauappcvgprlemladdru  7215  cauappcvgprlemladdrl  7216  cauappcvgprlem1  7218  caucvgprlemnkj  7225  caucvgprlemnbj  7226  caucvgprlemlol  7229  caucvgprlemloc  7234  caucvgprlemladdfu  7236  caucvgprlemladdrl  7237  caucvgprprlemnkltj  7248  caucvgprprlemnbj  7252  caucvgprprlemmu  7254  caucvgprprlemlol  7257  caucvgprprlemloc  7262  caucvgprprlemexbt  7265  caucvgprprlemexb  7266  caucvgprprlemaddq  7267  lttrsr  7308  ltsosr  7310  prsrlt  7332  caucvgsrlemoffcau  7343  caucvgsrlemoffgt1  7344  caucvgsrlemoffres  7345  caucvgsr  7347