Theorem caovord2d 6065

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 6064	. 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 6052	. . 3 |- ( ph -> ( C F A ) = ( A F C ) )
8	6, 4, 3	caovcomd 6052	. . 3 |- ( ph -> ( C F B ) = ( B F C ) )
9	7, 8	breq12d 4031	. 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 980   = wceq 1364   e. wcel 2160   class class class wbr 4018  (class class class)co 5895

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5898

This theorem is referenced by:  caovord3d  6066  genplt2i  7538  addnqprllem  7555  addnqprulem  7556  mulnqprl  7596  mulnqpru  7597  distrlem4prl  7612  distrlem4pru  7613  1idprl  7618  1idpru  7619  ltexprlemdisj  7634  ltexprlemloc  7635  ltexprlemfl  7637  ltexprlemfu  7639  prplnqu  7648  recexprlem1ssl  7661  recexprlem1ssu  7662  aptiprleml  7667  aptiprlemu  7668  caucvgprlemcanl  7672  cauappcvgprlemlol  7675  cauappcvgprlemloc  7680  cauappcvgprlemladdfu  7682  cauappcvgprlemladdru  7684  cauappcvgprlemladdrl  7685  cauappcvgprlem1  7687  caucvgprlemnkj  7694  caucvgprlemnbj  7695  caucvgprlemlol  7698  caucvgprlemloc  7703  caucvgprlemladdfu  7705  caucvgprlemladdrl  7706  caucvgprprlemnkltj  7717  caucvgprprlemnbj  7721  caucvgprprlemmu  7723  caucvgprprlemlol  7726  caucvgprprlemloc  7731  caucvgprprlemexbt  7734  caucvgprprlemexb  7735  caucvgprprlemaddq  7736  lttrsr  7790  ltsosr  7792  prsrlt  7815  caucvgsrlemoffcau  7826  caucvgsrlemoffgt1  7827  caucvgsrlemoffres  7828  caucvgsr  7830