Theorem reusv2lem5 5377

Description: Lemma for reusv2 5378. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Assertion

Ref	Expression
reusv2lem5	⊢ ((∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ ∅) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 = 𝐶))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶

Allowed substitution hint:   𝐶(𝑦)

Proof of Theorem reusv2lem5

Step	Hyp	Ref	Expression
1		tru 1544	. . . . . . . . 9 ⊢ ⊤
2		biimt 360	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝐴 ∧ ⊤) → (𝑥 = 𝐶 ↔ ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
3	1, 2	mpan2 691	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → (𝑥 = 𝐶 ↔ ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
4		ibar 528	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → (𝑥 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
5	3, 4	bitr3d 281	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → (((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝐶 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
6		eleq1 2823	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
7	6	pm5.32ri 575	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶) ↔ (𝐶 ∈ 𝐴 ∧ 𝑥 = 𝐶))
8	5, 7	bitr4di 289	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → (((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
9	8	ralimi 3074	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 (((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
10		ralbi 3093	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 (((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶)) → (∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
11	9, 10	syl 17	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 → (∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
12	11	eubidv 2586	. . 3 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 → (∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
13		r19.28zv 4481	. . . 4 ⊢ (𝐵 ≠ ∅ → (∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑥 = 𝐶)))
14	13	eubidv 2586	. . 3 ⊢ (𝐵 ≠ ∅ → (∃!𝑥∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑥 = 𝐶)))
15	12, 14	sylan9bb 509	. 2 ⊢ ((∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ ∅) → (∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑥 = 𝐶)))
16	1	biantrur 530	. . . . 5 ⊢ (𝑥 = 𝐶 ↔ (⊤ ∧ 𝑥 = 𝐶))
17	16	rexbii 3084	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃𝑦 ∈ 𝐵 (⊤ ∧ 𝑥 = 𝐶))
18	17	reubii 3373	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (⊤ ∧ 𝑥 = 𝐶))
19		reusv2lem4 5376	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (⊤ ∧ 𝑥 = 𝐶) ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶))
20	18, 19	bitri 275	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶))
21		df-reu 3365	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑥 = 𝐶))
22	15, 20, 21	3bitr4g 314	1 ⊢ ((∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ ∅) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  ∃!weu 2568   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  ∃!wreu 3362  ∅c0 4313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-nul 5281  ax-pow 5340

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-nul 4314

This theorem is referenced by:  reusv2  5378