Theorem caovord2d 6202

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 6201	. 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 6189	. . 3 |- ( ph -> ( C F A ) = ( A F C ) )
8	6, 4, 3	caovcomd 6189	. . 3 |- ( ph -> ( C F B ) = ( B F C ) )
9	7, 8	breq12d 4106	. 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 2202   class class class wbr 4093  (class class class)co 6028

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 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  caovord3d  6203  genplt2i  7790  addnqprllem  7807  addnqprulem  7808  mulnqprl  7848  mulnqpru  7849  distrlem4prl  7864  distrlem4pru  7865  1idprl  7870  1idpru  7871  ltexprlemdisj  7886  ltexprlemloc  7887  ltexprlemfl  7889  ltexprlemfu  7891  prplnqu  7900  recexprlem1ssl  7913  recexprlem1ssu  7914  aptiprleml  7919  aptiprlemu  7920  caucvgprlemcanl  7924  cauappcvgprlemlol  7927  cauappcvgprlemloc  7932  cauappcvgprlemladdfu  7934  cauappcvgprlemladdru  7936  cauappcvgprlemladdrl  7937  cauappcvgprlem1  7939  caucvgprlemnkj  7946  caucvgprlemnbj  7947  caucvgprlemlol  7950  caucvgprlemloc  7955  caucvgprlemladdfu  7957  caucvgprlemladdrl  7958  caucvgprprlemnkltj  7969  caucvgprprlemnbj  7973  caucvgprprlemmu  7975  caucvgprprlemlol  7978  caucvgprprlemloc  7983  caucvgprprlemexbt  7986  caucvgprprlemexb  7987  caucvgprprlemaddq  7988  lttrsr  8042  ltsosr  8044  prsrlt  8067  caucvgsrlemoffcau  8078  caucvgsrlemoffgt1  8079  caucvgsrlemoffres  8080  caucvgsr  8082