Theorem 2exopprim 44977

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 44524	. . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → {𝑎, 𝑏} = {𝐴, 𝐵}))
2	1	el2v 3440	. . . . 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 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 1539  ∃wex 1782  Vcvv 3432  {cpr 4563  ⟨cop 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568

This theorem is referenced by: (None)