Theorem 2exopprim 45953

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 45500	. . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → {𝑎, 𝑏} = {𝐴, 𝐵}))
2	1	el2v 3480	. . . . 5 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → {𝑎, 𝑏} = {𝐴, 𝐵})
3	2	eqcomd 2737	. . . 4 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → {𝐴, 𝐵} = {𝑎, 𝑏})
4	3	eqcoms 2739	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ → {𝐴, 𝐵} = {𝑎, 𝑏})
5	4	anim1i 615	. 2 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑))
6	5	2eximi 1838	1 ⊢ (∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781  Vcvv 3472  {cpr 4623  ⟨cop 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5291  ax-nul 5298  ax-pr 5419

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628

This theorem is referenced by: (None)