Theorem caovord3d 5671

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 3767	. 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 5670	. . 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 5669	. . 3 |- ( ph -> ( D R B <-> ( C F D ) R ( C F B ) ) )
10	7, 9	bibi12d 224	. 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 145	1 |- ( ph -> ( ( A F B ) = ( C F D ) -> ( A R C <-> D R B ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   e. wcel 1393   class class class wbr 3764  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  ordpipqqs  6472  ltsrprg  6832