Theorem 2exopprim 44865

Description: The existence of an ordered pair fulfilling a wff implies the existence of an unordered pair fulfilling the wff. (Contributed by AV, 29-Jul-2023.)

Assertion

Ref	Expression
2exopprim	⊢ (∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑))

Proof of Theorem 2exopprim

Step	Hyp	Ref	Expression
1		oppr 44411	. . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → {𝑎, 𝑏} = {𝐴, 𝐵}))
2	1	el2v 3430	. . . . 5 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → {𝑎, 𝑏} = {𝐴, 𝐵})
3	2	eqcomd 2744	. . . 4 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → {𝐴, 𝐵} = {𝑎, 𝑏})
4	3	eqcoms 2746	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ → {𝐴, 𝐵} = {𝑎, 𝑏})
5	4	anim1i 614	. 2 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑))
6	5	2eximi 1839	1 ⊢ (∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1783  Vcvv 3422  {cpr 4560  ⟨cop 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565

This theorem is referenced by: (None)