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Theorem oteqex2 5492
Description: Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 26-Apr-2015.)
Assertion
Ref Expression
oteqex2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → (𝐶 ∈ V ↔ 𝑇 ∈ V))

Proof of Theorem oteqex2
StepHypRef Expression
1 opeqex 5491 . 2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ V) ↔ (⟨𝑅, 𝑆⟩ ∈ V ∧ 𝑇 ∈ V)))
2 opex 5457 . . 3 𝐴, 𝐵⟩ ∈ V
32biantrur 530 . 2 (𝐶 ∈ V ↔ (⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ V))
4 opex 5457 . . 3 𝑅, 𝑆⟩ ∈ V
54biantrur 530 . 2 (𝑇 ∈ V ↔ (⟨𝑅, 𝑆⟩ ∈ V ∧ 𝑇 ∈ V))
61, 3, 53bitr4g 314 1 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → (𝐶 ∈ V ↔ 𝑇 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  cop 4629
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630
This theorem is referenced by:  oteqex  5493
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