Theorem reu3op 6281

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

Hypothesis

Ref	Expression
reu3op.a	⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
reu3op	⊢ (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 𝜒 ∧ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))

Distinct variable groups:   𝑋,𝑎,𝑏,𝑝,𝑥,𝑦   𝑌,𝑎,𝑏,𝑝,𝑥,𝑦   𝜓,𝑎,𝑏,𝑥,𝑦   𝜒,𝑝

Allowed substitution hints:   𝜓(𝑝)   𝜒(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem reu3op

Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reu3 3692	. 2 ⊢ (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ∧ ∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = 𝑞)))
2		reu3op.a	. . . 4 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓 ↔ 𝜒))
3	2	rexxp 5816	. . 3 ⊢ (∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 𝜒)
4		eqeq2 2776	. . . . . . 7 ⊢ (𝑞 = ⟨𝑥, 𝑦⟩ → (𝑝 = 𝑞 ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
5	4	imbi2d 342	. . . . . 6 ⊢ (𝑞 = ⟨𝑥, 𝑦⟩ → ((𝜓 → 𝑝 = 𝑞) ↔ (𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩)))
6	5	ralbidv 3187	. . . . 5 ⊢ (𝑞 = ⟨𝑥, 𝑦⟩ → (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = 𝑞) ↔ ∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩)))
7	6	rexxp 5816	. . . 4 ⊢ (∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = 𝑞) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩))
8		eqeq1 2768	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩))
9	2, 8	imbi12d 346	. . . . . . 7 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩) ↔ (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)))
10	9	ralxp 5815	. . . . . 6 ⊢ (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩))
11		eqcom 2771	. . . . . . . . 9 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)
12	11	a1i 11	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑌)) → (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))
13	12	imbi2d 342	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑌)) → ((𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩) ↔ (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
14	13	2ralbidva 3226	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
15	10, 14	bitrid 285	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
16	15	2rexbiia 3225	. . . 4 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))
17	7, 16	bitri 277	. . 3 ⊢ (∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = 𝑞) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))
18	3, 17	anbi12i 637	. 2 ⊢ ((∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ∧ ∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = 𝑞)) ↔ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 𝜒 ∧ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
19	1, 18	bitri 277	1 ⊢ (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 𝜒 ∧ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088  ∃!wreu 3367  ⟨cop 4590   × cxp 5647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-iun 4953  df-opab 5165  df-xp 5655  df-rel 5656

This theorem is referenced by:  opreu2reurex  6283