Theorem caovord2d 6262

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 6261	. 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 6249	. . 3 |- ( ph -> ( C F A ) = ( A F C ) )
8	6, 4, 3	caovcomd 6249	. . 3 |- ( ph -> ( C F B ) = ( B F C ) )
9	7, 8	breq12d 4144	. 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4131  (class class class)co 6088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-iota 5340  df-fv 5388  df-ov 6091

This theorem is used by:  caovord3d  6263  genplt2i  7883  addnqprllem  7900  addnqprulem  7901  mulnqprl  7941  mulnqpru  7942  distrlem4prl  7957  distrlem4pru  7958  1idprl  7963  1idpru  7964  ltexprlemdisj  7979  ltexprlemloc  7980  ltexprlemfl  7982  ltexprlemfu  7984  prplnqu  7993  recexprlem1ssl  8006  recexprlem1ssu  8007  aptiprleml  8012  aptiprlemu  8013  caucvgprlemcanl  8017  cauappcvgprlemlol  8020  cauappcvgprlemloc  8025  cauappcvgprlemladdfu  8027  cauappcvgprlemladdru  8029  cauappcvgprlemladdrl  8030  cauappcvgprlem1  8032  caucvgprlemnkj  8039  caucvgprlemnbj  8040  caucvgprlemlol  8043  caucvgprlemloc  8048  caucvgprlemladdfu  8050  caucvgprlemladdrl  8051  caucvgprprlemnkltj  8062  caucvgprprlemnbj  8066  caucvgprprlemmu  8068  caucvgprprlemlol  8071  caucvgprprlemloc  8076  caucvgprprlemexbt  8079  caucvgprprlemexb  8080  caucvgprprlemaddq  8081  lttrsr  8135  ltsosr  8137  prsrlt  8160  caucvgsrlemoffcau  8171  caucvgsrlemoffgt1  8172  caucvgsrlemoffres  8173  caucvgsr  8175