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

Theorem opthg 4197
 Description: Ordered pair theorem. 𝐶 and 𝐷 are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opthg ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Proof of Theorem opthg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3741 . . . 4 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21eqeq1d 2166 . . 3 (𝑥 = 𝐴 → (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩))
3 eqeq1 2164 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐶𝐴 = 𝐶))
43anbi1d 461 . . 3 (𝑥 = 𝐴 → ((𝑥 = 𝐶𝑦 = 𝐷) ↔ (𝐴 = 𝐶𝑦 = 𝐷)))
52, 4bibi12d 234 . 2 (𝑥 = 𝐴 → ((⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶𝑦 = 𝐷)) ↔ (⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝑦 = 𝐷))))
6 opeq2 3742 . . . 4 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
76eqeq1d 2166 . . 3 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩))
8 eqeq1 2164 . . . 4 (𝑦 = 𝐵 → (𝑦 = 𝐷𝐵 = 𝐷))
98anbi2d 460 . . 3 (𝑦 = 𝐵 → ((𝐴 = 𝐶𝑦 = 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
107, 9bibi12d 234 . 2 (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝑦 = 𝐷)) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))))
11 vex 2715 . . 3 𝑥 ∈ V
12 vex 2715 . . 3 𝑦 ∈ V
1311, 12opth 4196 . 2 (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶𝑦 = 𝐷))
145, 10, 13vtocl2g 2776 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335   ∈ wcel 2128  ⟨cop 3563 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569 This theorem is referenced by:  opthg2  4198  xpopth  6118  eqop  6119  inl11  6999  preqlu  7375  cauappcvgprlemladd  7561  elrealeu  7732  qnumdenbi  12046  crth  12076
 Copyright terms: Public domain W3C validator