Theorem reu3op 6292

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 3715	. 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 5833	. . 3 ⊢ (∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 𝜒)
4		eqeq2 2746	. . . . . . 7 ⊢ (𝑞 = ⟨𝑥, 𝑦⟩ → (𝑝 = 𝑞 ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
5	4	imbi2d 340	. . . . . 6 ⊢ (𝑞 = ⟨𝑥, 𝑦⟩ → ((𝜓 → 𝑝 = 𝑞) ↔ (𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩)))
6	5	ralbidv 3165	. . . . 5 ⊢ (𝑞 = ⟨𝑥, 𝑦⟩ → (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = 𝑞) ↔ ∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩)))
7	6	rexxp 5833	. . . 4 ⊢ (∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = 𝑞) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩))
8		eqeq1 2738	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩))
9	2, 8	imbi12d 344	. . . . . . 7 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩) ↔ (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)))
10	9	ralxp 5832	. . . . . 6 ⊢ (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩))
11		eqcom 2741	. . . . . . . . 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 340	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑌)) → ((𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩) ↔ (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
14	13	2ralbidva 3206	. . . . . 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 283	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
16	15	2rexbiia 3205	. . . 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 275	. . 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 628	. 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 275	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 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059  ∃!wreu 3361  ⟨cop 4612   × cxp 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-iun 4973  df-opab 5186  df-xp 5671  df-rel 5672

This theorem is referenced by:  opreu2reurex  6294