Theorem vtocl3ga 3569

Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.) Reduce axiom usage. (Revised by Gino Giotto, 3-Oct-2024.)

Hypotheses

Ref	Expression
vtocl3ga.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl3ga.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
vtocl3ga.3	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
vtocl3ga.4	⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑)

Assertion

Ref	Expression
vtocl3ga	⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝐷,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜓,𝑥   𝜒,𝑦   𝜃,𝑧

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

Proof of Theorem vtocl3ga

Step	Hyp	Ref	Expression
1		eleq1 2821	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐷 ↔ 𝐴 ∈ 𝐷))
2	1	3anbi1d 1440	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) ↔ (𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆)))
3		vtocl3ga.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	2, 3	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑) ↔ ((𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜓)))
5		eleq1 2821	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑅 ↔ 𝐵 ∈ 𝑅))
6	5	3anbi2d 1441	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆)))
7		vtocl3ga.2	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
8	6, 7	imbi12d 344	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜓) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜒)))
9		eleq1 2821	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ 𝑆 ↔ 𝐶 ∈ 𝑆))
10	9	3anbi3d 1442	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆)))
11		vtocl3ga.3	. . . 4 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
12	10, 11	imbi12d 344	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜒) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃)))
13		vtocl3ga.4	. . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑)
14	4, 8, 12, 13	vtocl3g 3563	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃))
15	14	pm2.43i 52	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476

This theorem is referenced by:  preq12bg  4854  poclOLD  5596  xpord3pred  8137  jensenlem2  26489  wrdt2ind  32112