Theorem caovord3d 5691

Description: Ordering law. (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 ) )
caovord3d.5	|- ( ph -> D e. S )

Assertion

Ref	Expression
caovord3d	|- ( ph -> ( ( A F B ) = ( C F D ) -> ( A R C <-> D R B ) ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   x, D, y, z   ph, x, y, z   x, F, y, z   x, R, y, z   x, S, y, z

Proof of Theorem caovord3d

Step	Hyp	Ref	Expression
1		breq1 3788	. 2 |- ( ( A F B ) = ( C F D ) -> ( ( A F B ) R ( C F B ) <-> ( C F D ) R ( C F B ) ) )
2		caovordg.1	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )
3		caovordd.2	. . . 4 |- ( ph -> A e. S )
4		caovordd.4	. . . 4 |- ( ph -> C e. S )
5		caovordd.3	. . . 4 |- ( ph -> B e. S )
6		caovord2d.com	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
7	2, 3, 4, 5, 6	caovord2d 5690	. . 3 |- ( ph -> ( A R C <-> ( A F B ) R ( C F B ) ) )
8		caovord3d.5	. . . 4 |- ( ph -> D e. S )
9	2, 8, 5, 4	caovordd 5689	. . 3 |- ( ph -> ( D R B <-> ( C F D ) R ( C F B ) ) )
10	7, 9	bibi12d 233	. 2 |- ( ph -> ( ( A R C <-> D R B ) <-> ( ( A F B ) R ( C F B ) <-> ( C F D ) R ( C F B ) ) ) )
11	1, 10	syl5ibr 154	1 |- ( ph -> ( ( A F B ) = ( C F D ) -> ( A R C <-> D R B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  ordpipqqs  6564  ltsrprg  6924