Theorem vtocl3ga 3540

Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.) Reduce axiom usage. (Revised by GG, 3-Oct-2024.) (Proof shortened by Wolf Lammen, 31-May-2025.)

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		vtocl3ga.3	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
2	1	imbi2d 343	. . . 4 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅) → 𝜒) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅) → 𝜃)))
3		vtocl3ga.1	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	3	imbi2d 343	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧 ∈ 𝑆 → 𝜑) ↔ (𝑧 ∈ 𝑆 → 𝜓)))
5		vtocl3ga.2	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
6	5	imbi2d 343	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ 𝑆 → 𝜓) ↔ (𝑧 ∈ 𝑆 → 𝜒)))
7		vtocl3ga.4	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑)
8	7	3expia 1139	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅) → (𝑧 ∈ 𝑆 → 𝜑))
9	4, 6, 8	vtocl2ga 3537	. . . . 5 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅) → (𝑧 ∈ 𝑆 → 𝜒))
10	9	com12 33	. . . 4 ⊢ (𝑧 ∈ 𝑆 → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅) → 𝜒))
11	2, 10	vtoclga 3536	. . 3 ⊢ (𝐶 ∈ 𝑆 → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅) → 𝜃))
12	11	impcom 413	. 2 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅) ∧ 𝐶 ∈ 𝑆) → 𝜃)
13	12	3impa 1127	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835

This theorem is used by:  xpord3pred  8169  jensenlem2  27315  wrdt2ind  33516