Theorem vtocl3g 3541

Description: Implicit substitution of a class for a setvar variable. Version of vtocl3gf 3539 with disjoint variable conditions instead of nonfreeness hypotheses, requiring fewer axioms. (Contributed by GG, 3-Oct-2024.)

Hypotheses

Ref	Expression
vtocl3g.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl3g.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
vtocl3g.3	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
vtocl3g.4	⊢ 𝜑

Assertion

Ref	Expression
vtocl3g	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑧,𝐴   𝑦,𝐵   𝑧,𝐵   𝑧,𝐶   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)   𝑉(𝑥, 𝑦, 𝑧)   𝑊(𝑥, 𝑦, 𝑧)   𝑋(𝑥, 𝑦, 𝑧)

Proof of Theorem vtocl3g

Step	Hyp	Ref	Expression
1		elex 3478	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vtocl3g.2	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
3	2	imbi2d 343	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒)))
4		vtocl3g.3	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
5	4	imbi2d 343	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴 ∈ V → 𝜒) ↔ (𝐴 ∈ V → 𝜃)))
6		vtocl3g.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7		vtocl3g.4	. . . . 5 ⊢ 𝜑
8	6, 7	vtoclg 3524	. . . 4 ⊢ (𝐴 ∈ V → 𝜓)
9	3, 5, 8	vtocl2g 3540	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ V → 𝜃))
10	1, 9	mpan9 516	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) → 𝜃)
11	10	3impb 1132	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459

This theorem is used by:  preq12bg  4820