Theorem reusv2lem4 5301

Description: Lemma for reusv2 5303. (Contributed by NM, 13-Dec-2012.)

Assertion

Ref	Expression
reusv2lem4	⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶))

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

Allowed substitution hints:   𝜑(𝑦)   𝐵(𝑦)   𝐶(𝑦)

Proof of Theorem reusv2lem4

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-reu 3145	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶)))
2		anass 471	. . . . . 6 ⊢ (((𝑦 ∈ 𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝜑)) ∧ 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ((𝐶 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 = 𝐶)))
3		rabid 3378	. . . . . . 7 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} ↔ (𝑦 ∈ 𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝜑)))
4	3	anbi1i 625	. . . . . 6 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} ∧ 𝑥 = 𝐶) ↔ ((𝑦 ∈ 𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝜑)) ∧ 𝑥 = 𝐶))
5		anass 471	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 = 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝑥 = 𝐶)))
6		eleq1 2900	. . . . . . . . . 10 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
7	6	anbi1d 631	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝐶 ∈ 𝐴 ∧ 𝜑)))
8	7	pm5.32ri 578	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 = 𝐶) ↔ ((𝐶 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 = 𝐶))
9	5, 8	bitr3i 279	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝑥 = 𝐶)) ↔ ((𝐶 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 = 𝐶))
10	9	anbi2i 624	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝑥 = 𝐶))) ↔ (𝑦 ∈ 𝐵 ∧ ((𝐶 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 = 𝐶)))
11	2, 4, 10	3bitr4ri 306	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝑥 = 𝐶))) ↔ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} ∧ 𝑥 = 𝐶))
12	11	rexbii2 3245	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝑥 = 𝐶)) ↔ ∃𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = 𝐶)
13		r19.42v 3350	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝑥 = 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶)))
14		nfrab1 3384	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}
15		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑧{𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}
16		nfv 1911	. . . . 5 ⊢ Ⅎ𝑧 𝑥 = 𝐶
17		nfcsb1v 3906	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶
18	17	nfeq2 2995	. . . . 5 ⊢ Ⅎ𝑦 𝑥 = ⦋𝑧 / 𝑦⦌𝐶
19		csbeq1a 3896	. . . . . 6 ⊢ (𝑦 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑦⦌𝐶)
20	19	eqeq2d 2832	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑥 = ⦋𝑧 / 𝑦⦌𝐶))
21	14, 15, 16, 18, 20	cbvrexfw 3438	. . . 4 ⊢ (∃𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = 𝐶 ↔ ∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
22	12, 13, 21	3bitr3i 303	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶)) ↔ ∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
23	22	eubii 2666	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶)) ↔ ∃!𝑥∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
24		elex 3512	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V)
25	24	ad2antrl 726	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝜑)) → 𝐶 ∈ V)
26	3, 25	sylbi 219	. . . . . 6 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝐶 ∈ V)
27	26	rgen 3148	. . . . 5 ⊢ ∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝐶 ∈ V
28		nfv 1911	. . . . . 6 ⊢ Ⅎ𝑧 𝐶 ∈ V
29	17	nfel1 2994	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶 ∈ V
30	19	eleq1d 2897	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐶 ∈ V ↔ ⦋𝑧 / 𝑦⦌𝐶 ∈ V))
31	14, 15, 28, 29, 30	cbvralfw 3437	. . . . 5 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝐶 ∈ V ↔ ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}⦋𝑧 / 𝑦⦌𝐶 ∈ V)
32	27, 31	mpbi 232	. . . 4 ⊢ ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}⦋𝑧 / 𝑦⦌𝐶 ∈ V
33		reusv2lem3 5300	. . . 4 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}⦋𝑧 / 𝑦⦌𝐶 ∈ V → (∃!𝑥∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃!𝑥∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶))
34	32, 33	ax-mp 5	. . 3 ⊢ (∃!𝑥∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃!𝑥∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
35		df-ral 3143	. . . . 5 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = ⦋𝑧 / 𝑦⦌𝐶))
36		nfv 1911	. . . . . 6 ⊢ Ⅎ𝑧(𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = 𝐶)
37	14	nfcri 2971	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}
38	37, 18	nfim 1893	. . . . . 6 ⊢ Ⅎ𝑦(𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
39		eleq1 2900	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} ↔ 𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}))
40	39, 20	imbi12d 347	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = 𝐶) ↔ (𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = ⦋𝑧 / 𝑦⦌𝐶)))
41	36, 38, 40	cbvalv1 2357	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = 𝐶) ↔ ∀𝑧(𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = ⦋𝑧 / 𝑦⦌𝐶))
42	3	imbi1i 352	. . . . . . . 8 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = 𝐶) ↔ ((𝑦 ∈ 𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝐶))
43		impexp 453	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶)))
44	42, 43	bitri 277	. . . . . . 7 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶)))
45	44	albii 1816	. . . . . 6 ⊢ (∀𝑦(𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = 𝐶) ↔ ∀𝑦(𝑦 ∈ 𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶)))
46		df-ral 3143	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶) ↔ ∀𝑦(𝑦 ∈ 𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶)))
47	45, 46	bitr4i 280	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶))
48	35, 41, 47	3bitr2i 301	. . . 4 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶))
49	48	eubii 2666	. . 3 ⊢ (∃!𝑥∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶))
50	34, 49	bitri 277	. 2 ⊢ (∃!𝑥∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶))
51	1, 23, 50	3bitri 299	1 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533   ∈ wcel 2110  ∃!weu 2649  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140  {crab 3142  Vcvv 3494  ⦋csb 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5209  ax-pow 5265

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-nul 4291

This theorem is referenced by:  reusv2lem5  5302