Theorem caovord3d 6117

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 4047	. 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 6116	. . 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 6115	. . 3 |- ( ph -> ( D R B <-> ( C F D ) R ( C F B ) ) )
10	7, 9	bibi12d 235	. 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	imbitrrid 156	1 |- ( ph -> ( ( A F B ) = ( C F D ) -> ( A R C <-> D R B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   class class class wbr 4044  (class class class)co 5944

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  ordpipqqs  7487  ltsrprg  7860