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Theorem opeqex 5500
Description: Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)
Assertion
Ref Expression
opeqex (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)))

Proof of Theorem opeqex
StepHypRef Expression
1 neeq1 3000 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ ⟨𝐶, 𝐷⟩ ≠ ∅))
2 opnz 5475 . 2 (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 opnz 5475 . 2 (⟨𝐶, 𝐷⟩ ≠ ∅ ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V))
41, 2, 33bitr3g 313 1 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wne 2937  Vcvv 3471  c0 4323  cop 4635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636
This theorem is referenced by:  oteqex2  5501  oteqex  5502
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