Theorem opreu2reurex 6194

Description: There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 24-Jun-2023.) (Revised by AV, 1-Jul-2023.)

Hypothesis

Ref	Expression
opreu2reurex.a	⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
opreu2reurex	⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑝   𝐵,𝑎,𝑏,𝑝   𝜑,𝑎,𝑏   𝜒,𝑝

Allowed substitution hints:   𝜑(𝑝)   𝜒(𝑎,𝑏)

Proof of Theorem opreu2reurex

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqcom 2746	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
2		vex 3434	. . . . . . . . 9 ⊢ 𝑎 ∈ V
3		vex 3434	. . . . . . . . 9 ⊢ 𝑏 ∈ V
4	2, 3	opth 5393	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))
5	1, 4	bitri 274	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ↔ (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))
6	5	imbi2i 335	. . . . . 6 ⊢ ((𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)))
7	6	a1i 11	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → ((𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))))
8	7	2ralbidva 3123	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))))
9	8	2rexbiia 3228	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)))
10	9	anbi2i 622	. 2 ⊢ ((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)) ↔ (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))))
11		opreu2reurex.a	. . 3 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))
12	11	reu3op 6192	. 2 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
13		2reu4 4462	. 2 ⊢ ((∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒) ↔ (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))))
14	10, 12, 13	3bitr4i 302	1 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1541   ∈ wcel 2109  ∀wral 3065  ∃wrex 3066  ∃!wreu 3067  ⟨cop 4572   × cxp 5586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-iun 4931  df-opab 5141  df-xp 5594  df-rel 5595

This theorem is referenced by:  opreu2reu  6195  2sqreuop  26591  2sqreuopnn  26592  2sqreuoplt  26593  2sqreuopltb  26594  2sqreuopnnlt  26595  2sqreuopnnltb  26596  opreu2reu1  30811