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Theorem opth2 5076
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 5075 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
41, 2, 3mp2an 672 1 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cop 4322
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 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323
This theorem is referenced by:  eqvinop  5082  opelxp  5286  fsn  6545  opiota  7378  canthwe  9675  ltresr  10163  mat1dimelbas  20495  fmucndlem  22315  hgt750lemb  31074  diblsmopel  36981  cdlemn7  37013  dihordlem7  37024  xihopellsmN  37064  dihopellsm  37065  dihpN  37146
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