Theorem vtocl3gaf 3593

Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.) (Proof shortened by Wolf Lammen, 31-May-2025.)

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.f	. . . 4 ⊢ Ⅎ𝑧𝐶
2		vtocl3gaf.c	. . . . . . 7 ⊢ Ⅎ𝑧𝐴
3	2	nfel1 2925	. . . . . 6 ⊢ Ⅎ𝑧 𝐴 ∈ 𝑅
4		vtocl3gaf.e	. . . . . . 7 ⊢ Ⅎ𝑧𝐵
5	4	nfel1 2925	. . . . . 6 ⊢ Ⅎ𝑧 𝐵 ∈ 𝑆
6	3, 5	nfan 1898	. . . . 5 ⊢ Ⅎ𝑧(𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)
7		vtocl3gaf.3	. . . . 5 ⊢ Ⅎ𝑧𝜃
8	6, 7	nfim 1895	. . . 4 ⊢ Ⅎ𝑧((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 𝜃)
9		vtocl3gaf.6	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
10	9	imbi2d 340	. . . 4 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 𝜒) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 𝜃)))
11		vtocl3gaf.a	. . . . . 6 ⊢ Ⅎ𝑥𝐴
12		vtocl3gaf.b	. . . . . 6 ⊢ Ⅎ𝑦𝐴
13		vtocl3gaf.d	. . . . . 6 ⊢ Ⅎ𝑦𝐵
14		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 ∈ 𝑇
15		vtocl3gaf.1	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
16	14, 15	nfim 1895	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 ∈ 𝑇 → 𝜓)
17		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑇
18		vtocl3gaf.2	. . . . . . 7 ⊢ Ⅎ𝑦𝜒
19	17, 18	nfim 1895	. . . . . 6 ⊢ Ⅎ𝑦(𝑧 ∈ 𝑇 → 𝜒)
20		vtocl3gaf.4	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
21	20	imbi2d 340	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧 ∈ 𝑇 → 𝜑) ↔ (𝑧 ∈ 𝑇 → 𝜓)))
22		vtocl3gaf.5	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
23	22	imbi2d 340	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ 𝑇 → 𝜓) ↔ (𝑧 ∈ 𝑇 → 𝜒)))
24		vtocl3gaf.7	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) → 𝜑)
25	24	3expia 1121	. . . . . 6 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑇 → 𝜑))
26	11, 12, 13, 16, 19, 21, 23, 25	vtocl2gaf 3591	. . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝑧 ∈ 𝑇 → 𝜒))
27	26	com12 32	. . . 4 ⊢ (𝑧 ∈ 𝑇 → ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 𝜒))
28	1, 8, 10, 27	vtoclgaf 3588	. . 3 ⊢ (𝐶 ∈ 𝑇 → ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 𝜃))
29	28	impcom 407	. 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑇) → 𝜃)
30	29	3impa 1110	1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  Ⅎwnfc 2893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-v 3490

This theorem is referenced by:  vtocl3gaOLDOLD  3597