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

Theorem opthg 5346
 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 4776 . . . 4 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21eqeq1d 2824 . . 3 (𝑥 = 𝐴 → (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩))
3 eqeq1 2826 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐶𝐴 = 𝐶))
43anbi1d 632 . . 3 (𝑥 = 𝐴 → ((𝑥 = 𝐶𝑦 = 𝐷) ↔ (𝐴 = 𝐶𝑦 = 𝐷)))
52, 4bibi12d 349 . 2 (𝑥 = 𝐴 → ((⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶𝑦 = 𝐷)) ↔ (⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝑦 = 𝐷))))
6 opeq2 4778 . . . 4 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
76eqeq1d 2824 . . 3 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩))
8 eqeq1 2826 . . . 4 (𝑦 = 𝐵 → (𝑦 = 𝐷𝐵 = 𝐷))
98anbi2d 631 . . 3 (𝑦 = 𝐵 → ((𝐴 = 𝐶𝑦 = 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
107, 9bibi12d 349 . 2 (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝑦 = 𝐷)) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))))
11 vex 3472 . . 3 𝑥 ∈ V
12 vex 3472 . . 3 𝑦 ∈ V
1311, 12opth 5345 . 2 (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶𝑦 = 𝐷))
145, 10, 13vtocl2g 3547 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ⟨cop 4545 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546 This theorem is referenced by:  opth1g  5347  opthg2  5348  opthneg  5350  otthg  5354  oteqex  5367  s111  13960  embedsetcestrclem  17398  symg2bas  18512  frgpnabllem1  18984  frgpnabllem2  18985  mat1dimbas  21075  linds2eq  30976  goeleq12bg  32670  opideq  35718  dvheveccl  38366  hoidmv1le  43172  oppr  43561  opprb  43562  prproropf1olem4  43962
 Copyright terms: Public domain W3C validator