Theorem eliin2f 45552

Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
eliin2f.1	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
eliin2f	⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem eliin2f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliin 4933	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
2	1	adantl 482	. 2 ⊢ ((𝐵 ≠ ∅ ∧ 𝐴 ∈ V) → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
3		prcnel 3458	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)
4	3	adantl 482	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)
5		n0 4288	. . . . . . . 8 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
6	5	birani 504	. . . . . . 7 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦 𝑦 ∈ 𝐵)
7		prcnel 3458	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)
8	7	a1d 25	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ V → (𝑦 ∈ 𝐵 → ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
9	8	adantl 482	. . . . . . . . 9 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦 ∈ 𝐵 → ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
10	9	ancld 555	. . . . . . . 8 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)))
11	10	eximdv 1924	. . . . . . 7 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (∃𝑦 𝑦 ∈ 𝐵 → ∃𝑦(𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)))
12	6, 11	mpd 15	. . . . . 6 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦(𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
13		df-rex 3065	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
14	12, 13	sylibr 235	. . . . 5 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦 ∈ 𝐵 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)
15		eliin2f.1	. . . . . 6 ⊢ Ⅎ𝑥𝐵
16		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑦𝐵
17		nfv 1921	. . . . . 6 ⊢ Ⅎ𝑦 ¬ 𝐴 ∈ 𝐶
18		nfcsb1v 3862	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
19	18	nfel2 2920	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶
20	19	nfn 1864	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶
21		csbeq1a 3852	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
22	21	eleq2d 2826	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
23	22	notbid 319	. . . . . 6 ⊢ (𝑥 = 𝑦 → (¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
24	15, 16, 17, 20, 23	cbvrexfw 3281	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 ¬ 𝐴 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)
25	14, 24	sylibr 235	. . . 4 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑥 ∈ 𝐵 ¬ 𝐴 ∈ 𝐶)
26		rexnal 3092	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 ¬ 𝐴 ∈ 𝐶 ↔ ¬ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
27	25, 26	sylib 219	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
28	4, 27	2falsed 377	. 2 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
29	2, 28	pm2.61dan 818	1 ⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∃wex 1786   ∈ wcel 2119  Ⅎwnfc 2887   ≠ wne 2935  ∀wral 3054  ∃wrex 3064  Vcvv 3432  ⦋csb 3838  ∅c0 4268  ∩ ciin 4929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-nul 4269  df-iin 4931

This theorem is referenced by:  eliin2  45564