Theorem oteqex2 5437

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 5436	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ V) ↔ (⟨𝑅, 𝑆⟩ ∈ V ∧ 𝑇 ∈ V)))
2		opex 5402	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3	2	biantrur 530	. 2 ⊢ (𝐶 ∈ V ↔ (⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ V))
4		opex 5402	. . 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 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580

This theorem is referenced by:  oteqex  5438