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Theorem oteqex2 5503
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 5502 . 2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ V) ↔ (⟨𝑅, 𝑆⟩ ∈ V ∧ 𝑇 ∈ V)))
2 opex 5468 . . 3 𝐴, 𝐵⟩ ∈ V
32biantrur 530 . 2 (𝐶 ∈ V ↔ (⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ V))
4 opex 5468 . . 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 206  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  cop 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632
This theorem is referenced by:  oteqex  5504
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