Theorem caovord3 6070

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 6069	. . . 4 |- ( B e. S -> ( A R C <-> ( A F B ) R ( C F B ) ) )
7	6	adantr 276	. . 3 |- ( ( B e. S /\ C e. S ) -> ( A R C <-> ( A F B ) R ( C F B ) ) )
8		breq1 4021	. . 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 462	. 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 6068	. . 3 |- ( C e. S -> ( D R B <-> ( C F D ) R ( C F B ) ) )
12	11	ad2antlr 489	. 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 191	1 |- ( ( ( B e. S /\ C e. S ) /\ ( A F B ) = ( C F D ) ) -> ( A R C <-> D R B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   _Vcvv 2752   class class class wbr 4018  (class class class)co 5896

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5899

This theorem is referenced by: (None)