Theorem reusv2lem4 5363

Description: Lemma for reusv2 5365. (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 3367	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶)))
2		anass 474	. . . . . 6 ⊢ (((𝑦 ∈ 𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝜑)) ∧ 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ((𝐶 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 = 𝐶)))
3		rabid 3433	. . . . . . 7 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} ↔ (𝑦 ∈ 𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝜑)))
4	3	anbi1i 636	. . . . . 6 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} ∧ 𝑥 = 𝐶) ↔ ((𝑦 ∈ 𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝜑)) ∧ 𝑥 = 𝐶))
5		anass 474	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 = 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝑥 = 𝐶)))
6		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
7	6	anbi1d 643	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝐶 ∈ 𝐴 ∧ 𝜑)))
8	7	pm5.32ri 586	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 = 𝐶) ↔ ((𝐶 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 = 𝐶))
9	5, 8	bitr3i 280	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝑥 = 𝐶)) ↔ ((𝐶 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 = 𝐶))
10	9	anbi2i 635	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝑥 = 𝐶))) ↔ (𝑦 ∈ 𝐵 ∧ ((𝐶 ∈ 𝐴 ∧ 𝜑) ∧ 𝑥 = 𝐶)))
11	2, 4, 10	3bitr4ri 307	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝑥 = 𝐶))) ↔ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} ∧ 𝑥 = 𝐶))
12	11	rexbii2 3106	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝑥 = 𝐶)) ↔ ∃𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = 𝐶)
13		r19.42v 3195	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝑥 = 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶)))
14		nfrab1 3432	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}
15		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑧{𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}
16		nfv 1947	. . . . 5 ⊢ Ⅎ𝑧 𝑥 = 𝐶
17		nfcsb1v 3871	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶
18	17	nfeq2 2940	. . . . 5 ⊢ Ⅎ𝑦 𝑥 = ⦋𝑧 / 𝑦⦌𝐶
19		csbeq1a 3861	. . . . . 6 ⊢ (𝑦 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑦⦌𝐶)
20	19	eqeq2d 2772	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑥 = ⦋𝑧 / 𝑦⦌𝐶))
21	14, 15, 16, 18, 20	cbvrexfw 3304	. . . 4 ⊢ (∃𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = 𝐶 ↔ ∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
22	12, 13, 21	3bitr3i 304	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶)) ↔ ∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
23	22	eubii 2611	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶)) ↔ ∃!𝑥∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
24		elex 3472	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V)
25	24	ad2antrl 741	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝜑)) → 𝐶 ∈ V)
26	3, 25	sylbi 220	. . . . . 6 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝐶 ∈ V)
27	26	rgen 3079	. . . . 5 ⊢ ∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝐶 ∈ V
28		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑧 𝐶 ∈ V
29	17	nfel1 2939	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶 ∈ V
30	19	eleq1d 2846	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐶 ∈ V ↔ ⦋𝑧 / 𝑦⦌𝐶 ∈ V))
31	14, 15, 28, 29, 30	cbvralfw 3303	. . . . 5 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝐶 ∈ V ↔ ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}⦋𝑧 / 𝑦⦌𝐶 ∈ V)
32	27, 31	mpbi 233	. . . 4 ⊢ ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}⦋𝑧 / 𝑦⦌𝐶 ∈ V
33		reusv2lem3 5362	. . . 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 3078	. . . . 5 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = ⦋𝑧 / 𝑦⦌𝐶))
36		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑧(𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = 𝐶)
37	14	nfcri 2915	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}
38	37, 18	nfim 1929	. . . . . 6 ⊢ Ⅎ𝑦(𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
39		eleq1 2849	. . . . . . 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 2371	. . . . 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 456	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶)))
44	42, 43	bitri 278	. . . . . . 7 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶)))
45	44	albii 1852	. . . . . 6 ⊢ (∀𝑦(𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = 𝐶) ↔ ∀𝑦(𝑦 ∈ 𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶)))
46		df-ral 3078	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶) ↔ ∀𝑦(𝑦 ∈ 𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶)))
47	45, 46	bitr4i 281	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)} → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶))
48	35, 41, 47	3bitr2i 302	. . . 4 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶))
49	48	eubii 2611	. . 3 ⊢ (∃!𝑥∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ (𝐶 ∈ 𝐴 ∧ 𝜑)}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶))
50	34, 49	bitri 278	. 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 300	1 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃!weu 2594  ∀wral 3077  ∃wrex 3087  ∃!wreu 3364  {crab 3413  Vcvv 3451  ⦋csb 3847

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-nul 5260  ax-pow 5327

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-nul 4280

This theorem is used by:  reusv2lem5  5364