Theorem 2exopprim 47512

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 47042	. . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → {𝑎, 𝑏} = {𝐴, 𝐵}))
2	1	el2v 3487	. . . . 5 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → {𝑎, 𝑏} = {𝐴, 𝐵})
3	2	eqcomd 2743	. . . 4 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → {𝐴, 𝐵} = {𝑎, 𝑏})
4	3	eqcoms 2745	. . 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 1836	1 ⊢ (∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779  Vcvv 3480  {cpr 4628  ⟨cop 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633

This theorem is referenced by: (None)