Theorem vtocl3 3567

Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Avoid ax-10 2139 and ax-11 2155. (Revised by GG, 20-Aug-2023.) (Proof shortened by Wolf Lammen, 23-Aug-2023.)

Hypotheses

Ref	Expression
vtocl3.1	⊢ 𝐴 ∈ V
vtocl3.2	⊢ 𝐵 ∈ V
vtocl3.3	⊢ 𝐶 ∈ V
vtocl3.4	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
vtocl3.5	⊢ 𝜑

Assertion

Ref	Expression
vtocl3	⊢ 𝜓

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝑧,𝐶,𝑦   𝜓,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑦,𝑧)   𝐵(𝑧)

Proof of Theorem vtocl3

Step	Hyp	Ref	Expression
1		vtocl3.3	. 2 ⊢ 𝐶 ∈ V
2		vtocl3.5	. . . 4 ⊢ 𝜑
3	2	a1i 11	. . 3 ⊢ (𝑧 = 𝐶 → 𝜑)
4		vtocl3.1	. . . 4 ⊢ 𝐴 ∈ V
5		vtocl3.2	. . . 4 ⊢ 𝐵 ∈ V
6		vtocl3.4	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
7	6	3expa 1117	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
8	7	pm5.74da 804	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑧 = 𝐶 → 𝜑) ↔ (𝑧 = 𝐶 → 𝜓)))
9	4, 5, 8, 3	vtocl2 3566	. . 3 ⊢ (𝑧 = 𝐶 → 𝜓)
10	3, 9	2thd 265	. 2 ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))
11	1, 10, 2	vtocl 3558	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  Vcvv 3478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1777  df-clel 2814

This theorem is referenced by: (None)