Theorem opeqex 5366

Description: Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)

Assertion

Ref	Expression
opeqex	⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)))

Proof of Theorem opeqex

Step	Hyp	Ref	Expression
1		neeq1 2994	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ ⟨𝐶, 𝐷⟩ ≠ ∅))
2		opnz 5342	. 2 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		opnz 5342	. 2 ⊢ (⟨𝐶, 𝐷⟩ ≠ ∅ ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V))
4	1, 2, 3	3bitr3g 316	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112   ≠ wne 2932  Vcvv 3398  ∅c0 4223  ⟨cop 4533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ne 2933  df-v 3400  df-dif 3856  df-un 3858  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534

This theorem is referenced by:  oteqex2  5367  oteqex  5368