MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opthg Structured version   Visualization version   GIF version

Theorem opthg 5074
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 4540 . . . 4 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21eqeq1d 2773 . . 3 (𝑥 = 𝐴 → (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩))
3 eqeq1 2775 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐶𝐴 = 𝐶))
43anbi1d 609 . . 3 (𝑥 = 𝐴 → ((𝑥 = 𝐶𝑦 = 𝐷) ↔ (𝐴 = 𝐶𝑦 = 𝐷)))
52, 4bibi12d 334 . 2 (𝑥 = 𝐴 → ((⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶𝑦 = 𝐷)) ↔ (⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝑦 = 𝐷))))
6 opeq2 4541 . . . 4 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
76eqeq1d 2773 . . 3 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩))
8 eqeq1 2775 . . . 4 (𝑦 = 𝐵 → (𝑦 = 𝐷𝐵 = 𝐷))
98anbi2d 608 . . 3 (𝑦 = 𝐵 → ((𝐴 = 𝐶𝑦 = 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
107, 9bibi12d 334 . 2 (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝑦 = 𝐷)) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))))
11 vex 3354 . . 3 𝑥 ∈ V
12 vex 3354 . . 3 𝑦 ∈ V
1311, 12opth 5073 . 2 (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶𝑦 = 𝐷))
145, 10, 13vtocl2g 3422 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  cop 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324
This theorem is referenced by:  opth1g  5075  opthg2  5076  opthneg  5078  otthg  5082  oteqex  5092  s111  13596  embedsetcestrclem  17006  symg2bas  18026  frgpnabllem1  18484  frgpnabllem2  18485  mat1dimbas  20497  opideq  34453  dvheveccl  36923  hoidmv1le  41329
  Copyright terms: Public domain W3C validator