Theorem caoprord3 3072

Description: Ordering law.

Hypotheses

Ref	Expression
caoprord.1	|- A e. V
caoprord.2	|- B e. V
caoprord.3	|- (z e. S -> (xRy <-> (zFx)R(zFy)))
caoprord2.3	|- C e. V
caoprord2.com	|- (xFy) = (yFx)
caoprord3.4	|- D e. V

Assertion

Ref	Expression
caoprord3	|- (((B e. S /\ C e. S) /\ (AFB) = (CFD)) -> (ARC <-> DRB))

Distinct variable group(s):   x,y,z,F   x,S,y,z   x,A,y,z   x,B,y,z   x,C,y,z   x,D,y,z   x,R,y,z

Proof of Theorem caoprord3

Step	Hyp	Ref	Expression
1		caoprord.1	. . . . 5 |- A e. V
2		caoprord2.3	. . . . 5 |- C e. V
3		caoprord.3	. . . . 5 |- (z e. S -> (xRy <-> (zFx)R(zFy)))
4		caoprord.2	. . . . 5 |- B e. V
5		caoprord2.com	. . . . 5 |- (xFy) = (yFx)
6	1, 2, 3, 4, 5	caoprord2 3071	. . . 4 |- (B e. S -> (ARC <-> (AFB)R(CFB)))
7	6	adantr 306	. . 3 |- ((B e. S /\ C e. S) -> (ARC <-> (AFB)R(CFB)))
8		breq1 2065	. . 3 |- ((AFB) = (CFD) -> ((AFB)R(CFB) <-> (CFD)R(CFB)))
9	7, 8	sylan9bb 418	. 2 |- (((B e. S /\ C e. S) /\ (AFB) = (CFD)) -> (ARC <-> (CFD)R(CFB)))
10		caoprord3.4	. . . 4 |- D e. V
11	10, 4, 3	caoprord 3070	. . 3 |- (C e. S -> (DRB <-> (CFD)R(CFB)))
12	11	ad2antlr 321	. 2 |- (((B e. S /\ C e. S) /\ (AFB) = (CFD)) -> (DRB <-> (CFD)R(CFB)))
13	9, 12	bitr4d 409	1 |- (((B e. S /\ C e. S) /\ (AFB) = (CFD)) -> (ARC <-> DRB))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348   class class class wbr 2054  (class class class)co 3001

This theorem is referenced by:  ordpipq 3850  genpnnp 3902  ltsrpr 3980

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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