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Theorem opth2 5485
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 5484 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
41, 2, 3mp2an 692 1 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633
This theorem is referenced by:  eqvinop  5492  opelxp  5721  fsn  7155  opiota  8084  canthwe  10691  ltresr  11180  mat1dimelbas  22477  fmucndlem  24300  hgt750lemb  34671  diblsmopel  41173  cdlemn7  41205  dihordlem7  41216  xihopellsmN  41256  dihopellsm  41257  dihpN  41338
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