Theorem vtocl3g 3501

Description: Implicit substitution of a class for a setvar variable. Version of vtocl3gf 3499 with disjoint variable conditions instead of nonfreeness hypotheses, requiring fewer axioms. (Contributed by Gino Giotto, 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 3440	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vtocl3g.2	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
3	2	imbi2d 340	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒)))
4		vtocl3g.3	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
5	4	imbi2d 340	. . . 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 3495	. . . 4 ⊢ (𝐴 ∈ V → 𝜓)
9	3, 5, 8	vtocl2g 3500	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ V → 𝜃))
10	1, 9	mpan9 506	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) → 𝜃)
11	10	3impb 1113	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424

This theorem is referenced by:  vtocl3ga  3507