ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caovord3 GIF version

Theorem caovord3 5702
Description: Ordering law. (Contributed by NM, 29-Feb-1996.)
Hypotheses
Ref Expression
caovord.1 𝐴 ∈ V
caovord.2 𝐵 ∈ V
caovord.3 (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
caovord2.3 𝐶 ∈ V
caovord2.com (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
caovord3.4 𝐷 ∈ V
Assertion
Ref Expression
caovord3 (((𝐵𝑆𝐶𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶𝐷𝑅𝐵))
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
StepHypRef Expression
1 caovord.1 . . . . 5 𝐴 ∈ V
2 caovord2.3 . . . . 5 𝐶 ∈ V
3 caovord.3 . . . . 5 (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
4 caovord.2 . . . . 5 𝐵 ∈ V
5 caovord2.com . . . . 5 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
61, 2, 3, 4, 5caovord2 5701 . . . 4 (𝐵𝑆 → (𝐴𝑅𝐶 ↔ (𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵)))
76adantr 265 . . 3 ((𝐵𝑆𝐶𝑆) → (𝐴𝑅𝐶 ↔ (𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵)))
8 breq1 3795 . . 3 ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → ((𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵) ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
97, 8sylan9bb 443 . 2 (((𝐵𝑆𝐶𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
10 caovord3.4 . . . 4 𝐷 ∈ V
1110, 4, 3caovord 5700 . . 3 (𝐶𝑆 → (𝐷𝑅𝐵 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
1211ad2antlr 466 . 2 (((𝐵𝑆𝐶𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐷𝑅𝐵 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
139, 12bitr4d 184 1 (((𝐵𝑆𝐶𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶𝐷𝑅𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  Vcvv 2574   class class class wbr 3792  (class class class)co 5540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator