Theorem opreu2reurex 6145

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 2828	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
2		vex 3497	. . . . . . . . 9 ⊢ 𝑎 ∈ V
3		vex 3497	. . . . . . . . 9 ⊢ 𝑏 ∈ V
4	2, 3	opth 5368	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))
5	1, 4	bitri 277	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ↔ (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))
6	5	imbi2i 338	. . . . . 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 3198	. . . 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 3298	. . 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 624	. 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 6143	. 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 4466	. 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 305	1 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140  ⟨cop 4573   × cxp 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-iun 4921  df-opab 5129  df-xp 5561  df-rel 5562

This theorem is referenced by:  opreu2reu  6146  2sqreuop  26038  2sqreuopnn  26039  2sqreuoplt  26040  2sqreuopltb  26041  2sqreuopnnlt  26042  2sqreuopnnltb  26043  opreu2reu1  30247