Theorem caoprord3 4072

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 groups:   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 4071	. . . 4 |- (B e. S -> (ARC <-> (AFB)R(CFB)))
7	6	adantr 391	. . 3 |- ((B e. S /\ C e. S) -> (ARC <-> (AFB)R(CFB)))
8		breq1 2635	. . 3 |- ((AFB) = (CFD) -> ((AFB)R(CFB) <-> (CFD)R(CFB)))
9	7, 8	sylan9bb 543	. 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 4070	. . 3 |- (C e. S -> (DRB <-> (CFD)R(CFB)))
12	11	ad2antlr 407	. 2 |- (((B e. S /\ C e. S) /\ (AFB) = (CFD)) -> (DRB <-> (CFD)R(CFB)))
13	9, 12	bitr4d 534	1 |- (((B e. S /\ C e. S) /\ (AFB) = (CFD)) -> (ARC <-> DRB))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 960   e. wcel 962  Vcvv 1818   class class class wbr 2632  (class class class)co 3977

This theorem is referenced by:  ordpipq 5069  genpnnp 5121  ltsrpr 5199

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-sep 2716  ax-pow 2756  ax-pr 2793

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-v 1819  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-uni 2516  df-br 2633  df-opab 2680  df-xp 3198  df-cnv 3200  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fv 3212  df-opr 3979

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