Theorem oteqex2 5431

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 5430	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ V) ↔ (⟨𝑅, 𝑆⟩ ∈ V ∧ 𝑇 ∈ V)))
2		opex 5397	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3	2	biantrur 531	. 2 ⊢ (𝐶 ∈ V ↔ (⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ V))
4		opex 5397	. . 3 ⊢ ⟨𝑅, 𝑆⟩ ∈ V
5	4	biantrur 531	. 2 ⊢ (𝑇 ∈ V ↔ (⟨𝑅, 𝑆⟩ ∈ V ∧ 𝑇 ∈ V))
6	1, 3, 5	3bitr4g 313	1 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → (𝐶 ∈ V ↔ 𝑇 ∈ V))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  Vcvv 3440  ⟨cop 4576

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pr 5366

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-v 3442  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577

This theorem is referenced by:  oteqex  5432