Theorem caovord3 5674

Description: Ordering law. (Contributed by NM, 29-Feb-1996.)

Hypotheses

Ref	Expression
caovord.1	|- A e. _V
caovord.2	|- B e. _V
caovord.3	|- ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) )
caovord2.3	|- C e. _V
caovord2.com	|- ( x F y ) = ( y F x )
caovord3.4	|- D e. _V

Assertion

Ref	Expression
caovord3	|- ( ( ( B e. S /\ C e. S ) /\ ( 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   x, F, y, z   x, R, y, z   x, S, y, z

Proof of Theorem caovord3

Step	Hyp	Ref	Expression
1		caovord.1	. . . . 5 |- A e. _V
2		caovord2.3	. . . . 5 |- C e. _V
3		caovord.3	. . . . 5 |- ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) )
4		caovord.2	. . . . 5 |- B e. _V
5		caovord2.com	. . . . 5 |- ( x F y ) = ( y F x )
6	1, 2, 3, 4, 5	caovord2 5673	. . . 4 |- ( B e. S -> ( A R C <-> ( A F B ) R ( C F B ) ) )
7	6	adantr 261	. . 3 |- ( ( B e. S /\ C e. S ) -> ( A R C <-> ( A F B ) R ( C F B ) ) )
8		breq1 3767	. . 3 |- ( ( A F B ) = ( C F D ) -> ( ( A F B ) R ( C F B ) <-> ( C F D ) R ( C F B ) ) )
9	7, 8	sylan9bb 435	. 2 |- ( ( ( B e. S /\ C e. S ) /\ ( A F B ) = ( C F D ) ) -> ( A R C <-> ( C F D ) R ( C F B ) ) )
10		caovord3.4	. . . 4 |- D e. _V
11	10, 4, 3	caovord 5672	. . 3 |- ( C e. S -> ( D R B <-> ( C F D ) R ( C F B ) ) )
12	11	ad2antlr 458	. 2 |- ( ( ( B e. S /\ C e. S ) /\ ( A F B ) = ( C F D ) ) -> ( D R B <-> ( C F D ) R ( C F B ) ) )
13	9, 12	bitr4d 180	1 |- ( ( ( B e. S /\ C e. S ) /\ ( A F B ) = ( C F D ) ) -> ( A R C <-> D R B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   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: (None)