Theorem opeqex 5508

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 3001	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ ⟨𝐶, 𝐷⟩ ≠ ∅))
2		opnz 5484	. 2 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		opnz 5484	. 2 ⊢ (⟨𝐶, 𝐷⟩ ≠ ∅ ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V))
4	1, 2, 3	3bitr3g 313	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  Vcvv 3478  ∅c0 4339  ⟨cop 4637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638

This theorem is referenced by:  oteqex2  5509  oteqex  5510