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Theorem opth2 5340
 Description: Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
Hypotheses
Ref Expression
opth2.1 𝐶 ∈ V
opth2.2 𝐷 ∈ V
Assertion
Ref Expression
opth2 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem opth2
StepHypRef Expression
1 opth2.1 . 2 𝐶 ∈ V
2 opth2.2 . 2 𝐷 ∈ V
3 opthg2 5339 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
41, 2, 3mp2an 691 1 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  Vcvv 3444  ⟨cop 4534 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 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 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535 This theorem is referenced by:  eqvinop  5346  opelxp  5559  fsn  6878  opiota  7743  canthwe  10066  ltresr  10555  mat1dimelbas  21080  fmucndlem  22901  hgt750lemb  32041  diblsmopel  38466  cdlemn7  38498  dihordlem7  38509  xihopellsmN  38549  dihopellsm  38550  dihpN  38631
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