Theorem caovord3 5818

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 5817	. . . 4 |- ( B e. S -> ( A R C <-> ( A F B ) R ( C F B ) ) )
7	6	adantr 270	. . 3 |- ( ( B e. S /\ C e. S ) -> ( A R C <-> ( A F B ) R ( C F B ) ) )
8		breq1 3848	. . 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 450	. 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 5816	. . 3 |- ( C e. S -> ( D R B <-> ( C F D ) R ( C F B ) ) )
12	11	ad2antlr 473	. 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 189	1 |- ( ( ( B e. S /\ C e. S ) /\ ( A F B ) = ( C F D ) ) -> ( A R C <-> D R B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   _Vcvv 2619   class class class wbr 3845  (class class class)co 5652

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655

This theorem is referenced by: (None)