Theorem reu3op 6295

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 3684	. 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 5818	. . 3 ⊢ (∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 𝜒)
4		eqeq2 2772	. . . . . . 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 3185	. . . . 5 ⊢ (𝑞 = ⟨𝑥, 𝑦⟩ → (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = 𝑞) ↔ ∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩)))
7	6	rexxp 5818	. . . 4 ⊢ (∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = 𝑞) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩))
8		eqeq1 2764	. . . . . . . 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 5817	. . . . . 6 ⊢ (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩))
11		eqcom 2767	. . . . . . . . 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 3224	. . . . . 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 286	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓 → 𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
16	15	2rexbiia 3223	. . . 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 278	. . 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 640	. 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 278	1 ⊢ (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 𝜒 ∧ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363  ⟨cop 4589   × cxp 5648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-iun 4952  df-opab 5167  df-xp 5656  df-rel 5657

This theorem is used by:  opreu2reurex  6297