Theorem caovord2d 6175

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 6174	. 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 6162	. . 3 |- ( ph -> ( C F A ) = ( A F C ) )
8	6, 4, 3	caovcomd 6162	. . 3 |- ( ph -> ( C F B ) = ( B F C ) )
9	7, 8	breq12d 4096	. 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 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6001

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004

This theorem is referenced by:  caovord3d  6176  genplt2i  7697  addnqprllem  7714  addnqprulem  7715  mulnqprl  7755  mulnqpru  7756  distrlem4prl  7771  distrlem4pru  7772  1idprl  7777  1idpru  7778  ltexprlemdisj  7793  ltexprlemloc  7794  ltexprlemfl  7796  ltexprlemfu  7798  prplnqu  7807  recexprlem1ssl  7820  recexprlem1ssu  7821  aptiprleml  7826  aptiprlemu  7827  caucvgprlemcanl  7831  cauappcvgprlemlol  7834  cauappcvgprlemloc  7839  cauappcvgprlemladdfu  7841  cauappcvgprlemladdru  7843  cauappcvgprlemladdrl  7844  cauappcvgprlem1  7846  caucvgprlemnkj  7853  caucvgprlemnbj  7854  caucvgprlemlol  7857  caucvgprlemloc  7862  caucvgprlemladdfu  7864  caucvgprlemladdrl  7865  caucvgprprlemnkltj  7876  caucvgprprlemnbj  7880  caucvgprprlemmu  7882  caucvgprprlemlol  7885  caucvgprprlemloc  7890  caucvgprprlemexbt  7893  caucvgprprlemexb  7894  caucvgprprlemaddq  7895  lttrsr  7949  ltsosr  7951  prsrlt  7974  caucvgsrlemoffcau  7985  caucvgsrlemoffgt1  7986  caucvgsrlemoffres  7987  caucvgsr  7989