Theorem reu3op 6143

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 3718	. 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 5713	. . 3 ⊢ (∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 𝜒)
4		eqeq2 2833	. . . . . . 7 ⊢ (𝑞 = ⟨𝑥, 𝑦⟩ → (𝑝 = 𝑞 ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
5	4	imbi2d 343	. . . . . 6 ⊢ (𝑞 = ⟨𝑥, 𝑦⟩ → ((𝜓 → 𝑝 = 𝑞) ↔ (𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩)))
6	5	ralbidv 3197	. . . . 5 ⊢ (𝑞 = ⟨𝑥, 𝑦⟩ → (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = 𝑞) ↔ ∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩)))
7	6	rexxp 5713	. . . 4 ⊢ (∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = 𝑞) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩))
8		eqeq1 2825	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩))
9	2, 8	imbi12d 347	. . . . . . 7 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩) ↔ (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)))
10	9	ralxp 5712	. . . . . 6 ⊢ (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩))
11		eqcom 2828	. . . . . . . . 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 343	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑌)) → ((𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩) ↔ (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
14	13	2ralbidva 3198	. . . . . 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	syl5bb 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 3298	. . . 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 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 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 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-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:  opreu2reurex  6145