Theorem oteqex2 5455

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

Step	Hyp	Ref	Expression
1		opeqex 5454	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ V) ↔ (⟨𝑅, 𝑆⟩ ∈ V ∧ 𝑇 ∈ V)))
2		opex 5419	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3	2	biantrur 530	. 2 ⊢ (𝐶 ∈ V ↔ (⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ V))
4		opex 5419	. . 3 ⊢ ⟨𝑅, 𝑆⟩ ∈ V
5	4	biantrur 530	. 2 ⊢ (𝑇 ∈ V ↔ (⟨𝑅, 𝑆⟩ ∈ V ∧ 𝑇 ∈ V))
6	1, 3, 5	3bitr4g 314	1 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → (𝐶 ∈ V ↔ 𝑇 ∈ V))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ⟨cop 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589

This theorem is referenced by:  oteqex  5456