Theorem 2exopprim 48163

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 47656	. . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → {𝑎, 𝑏} = {𝐴, 𝐵}))
2	1	el2v 3470	. . . . 5 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → {𝑎, 𝑏} = {𝐴, 𝐵})
3	2	eqcomd 2775	. . . 4 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → {𝐴, 𝐵} = {𝑎, 𝑏})
4	3	eqcoms 2777	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ → {𝐴, 𝐵} = {𝑎, 𝑏})
5	4	anim1i 626	. 2 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑))
6	5	2eximi 1863	1 ⊢ (∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567  ∃wex 1806  Vcvv 3463  {cpr 4596  ⟨cop 4600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601

This theorem is referenced by: (None)