Theorem vtocl3gaf 3516

Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)

Hypotheses

Ref	Expression
vtocl3gaf.a	⊢ Ⅎ𝑥𝐴
vtocl3gaf.b	⊢ Ⅎ𝑦𝐴
vtocl3gaf.c	⊢ Ⅎ𝑧𝐴
vtocl3gaf.d	⊢ Ⅎ𝑦𝐵
vtocl3gaf.e	⊢ Ⅎ𝑧𝐵
vtocl3gaf.f	⊢ Ⅎ𝑧𝐶
vtocl3gaf.1	⊢ Ⅎ𝑥𝜓
vtocl3gaf.2	⊢ Ⅎ𝑦𝜒
vtocl3gaf.3	⊢ Ⅎ𝑧𝜃
vtocl3gaf.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl3gaf.5	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
vtocl3gaf.6	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
vtocl3gaf.7	⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) → 𝜑)

Assertion

Ref	Expression
vtocl3gaf	⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝜃)

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)   𝜃(𝑥,𝑦,𝑧)   𝐴(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem vtocl3gaf

Step	Hyp	Ref	Expression
1		vtocl3gaf.a	. . 3 ⊢ Ⅎ𝑥𝐴
2		vtocl3gaf.b	. . 3 ⊢ Ⅎ𝑦𝐴
3		vtocl3gaf.c	. . 3 ⊢ Ⅎ𝑧𝐴
4		vtocl3gaf.d	. . 3 ⊢ Ⅎ𝑦𝐵
5		vtocl3gaf.e	. . 3 ⊢ Ⅎ𝑧𝐵
6		vtocl3gaf.f	. . 3 ⊢ Ⅎ𝑧𝐶
7	1	nfel1 2923	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑅
8		nfv 1917	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑆
9		nfv 1917	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝑇
10	7, 8, 9	nf3an 1904	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇)
11		vtocl3gaf.1	. . . 4 ⊢ Ⅎ𝑥𝜓
12	10, 11	nfim 1899	. . 3 ⊢ Ⅎ𝑥((𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) → 𝜓)
13	2	nfel1 2923	. . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ 𝑅
14	4	nfel1 2923	. . . . 5 ⊢ Ⅎ𝑦 𝐵 ∈ 𝑆
15		nfv 1917	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑇
16	13, 14, 15	nf3an 1904	. . . 4 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇)
17		vtocl3gaf.2	. . . 4 ⊢ Ⅎ𝑦𝜒
18	16, 17	nfim 1899	. . 3 ⊢ Ⅎ𝑦((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) → 𝜒)
19	3	nfel1 2923	. . . . 5 ⊢ Ⅎ𝑧 𝐴 ∈ 𝑅
20	5	nfel1 2923	. . . . 5 ⊢ Ⅎ𝑧 𝐵 ∈ 𝑆
21	6	nfel1 2923	. . . . 5 ⊢ Ⅎ𝑧 𝐶 ∈ 𝑇
22	19, 20, 21	nf3an 1904	. . . 4 ⊢ Ⅎ𝑧(𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇)
23		vtocl3gaf.3	. . . 4 ⊢ Ⅎ𝑧𝜃
24	22, 23	nfim 1899	. . 3 ⊢ Ⅎ𝑧((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝜃)
25		eleq1 2826	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑅 ↔ 𝐴 ∈ 𝑅))
26	25	3anbi1d 1439	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) ↔ (𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇)))
27		vtocl3gaf.4	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
28	26, 27	imbi12d 345	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) → 𝜑) ↔ ((𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) → 𝜓)))
29		eleq1 2826	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆))
30	29	3anbi2d 1440	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇)))
31		vtocl3gaf.5	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
32	30, 31	imbi12d 345	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) → 𝜓) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) → 𝜒)))
33		eleq1 2826	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ 𝑇 ↔ 𝐶 ∈ 𝑇))
34	33	3anbi3d 1441	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇)))
35		vtocl3gaf.6	. . . 4 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
36	34, 35	imbi12d 345	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) → 𝜒) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝜃)))
37		vtocl3gaf.7	. . 3 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) → 𝜑)
38	1, 2, 3, 4, 5, 6, 12, 18, 24, 28, 32, 36, 37	vtocl3gf 3509	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝜃))
39	38	pm2.43i 52	1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1086   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2106  Ⅎwnfc 2887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-v 3434

This theorem is referenced by:  vtocl3gaOLD  3518