Theorem caovord3 6847

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 6846	. . . 4 |- ( B e. S -> ( A R C <-> ( A F B ) R ( C F B ) ) )
7	6	adantr 481	. . 3 |- ( ( B e. S /\ C e. S ) -> ( A R C <-> ( A F B ) R ( C F B ) ) )
8		breq1 4656	. . 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 736	. 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 6845	. . 3 |- ( C e. S -> ( D R B <-> ( C F D ) R ( C F B ) ) )
12	11	ad2antlr 763	. 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 271	1 |- ( ( ( B e. S /\ C e. S ) /\ ( A F B ) = ( C F D ) ) -> ( A R C <-> D R B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  genpnnp  9827  ltsrpr  9898