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Theorem opthg 5479
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 4875 . . . 4 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21eqeq1d 2727 . . 3 (𝑥 = 𝐴 → (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩))
3 eqeq1 2729 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐶𝐴 = 𝐶))
43anbi1d 629 . . 3 (𝑥 = 𝐴 → ((𝑥 = 𝐶𝑦 = 𝐷) ↔ (𝐴 = 𝐶𝑦 = 𝐷)))
52, 4bibi12d 344 . 2 (𝑥 = 𝐴 → ((⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶𝑦 = 𝐷)) ↔ (⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝑦 = 𝐷))))
6 opeq2 4876 . . . 4 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
76eqeq1d 2727 . . 3 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩))
8 eqeq1 2729 . . . 4 (𝑦 = 𝐵 → (𝑦 = 𝐷𝐵 = 𝐷))
98anbi2d 628 . . 3 (𝑦 = 𝐵 → ((𝐴 = 𝐶𝑦 = 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
107, 9bibi12d 344 . 2 (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝑦 = 𝐷)) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))))
11 vex 3465 . . 3 𝑥 ∈ V
12 vex 3465 . . 3 𝑦 ∈ V
1311, 12opth 5478 . 2 (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶𝑦 = 𝐷))
145, 10, 13vtocl2g 3553 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  cop 4636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637
This theorem is referenced by:  opth1g  5480  opthg2  5481  opthneg  5483  otthg  5487  oteqex  5502  s111  14601  embedsetcestrclem  18151  symg2bas  19359  frgpnabllem1  19840  frgpnabllem2  19841  mat1dimbas  22418  linds2eq  33193  goeleq12bg  35090  opideq  37945  dvheveccl  40715  hoidmv1le  46120  oppr  46550  opprb  46551  fsetsnf1  46572  prproropf1olem4  46983
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