Theorem caoprord3 4991

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 4990	. . . 4 |- (B e. S -> (ARC <-> (AFB)R(CFB)))
7	6	adantr 425	. . 3 |- ((B e. S /\ C e. S) -> (ARC <-> (AFB)R(CFB)))
8		breq1 3341	. . 3 |- ((AFB) = (CFD) -> ((AFB)R(CFB) <-> (CFD)R(CFB)))
9	7, 8	sylan9bb 599	. 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 4989	. . 3 |- (C e. S -> (DRB <-> (CFD)R(CFB)))
12	11	ad2antlr 441	. 2 |- (((B e. S /\ C e. S) /\ (AFB) = (CFD)) -> (DRB <-> (CFD)R(CFB)))
13	9, 12	bitr4d 590	1 |- (((B e. S /\ C e. S) /\ (AFB) = (CFD)) -> (ARC <-> DRB))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   class class class wbr 3338  (class class class)co 4884

This theorem is referenced by:  ordpipq 6208  genpnnp 6260  ltsrpr 6338

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886

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