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Theorem vtocl3 3538
 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 2145 and ax-11 2161. (Revised by Gino Giotto, 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 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
vtocl3.5 𝜑
Assertion
Ref Expression
vtocl3 𝜓
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜓,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem vtocl3
StepHypRef Expression
1 vtocl3.3 . 2 𝐶 ∈ V
2 vtocl3.5 . . . 4 𝜑
32a1i 11 . . 3 (𝑧 = 𝐶𝜑)
4 vtocl3.1 . . . 4 𝐴 ∈ V
5 vtocl3.2 . . . 4 𝐵 ∈ V
6 vtocl3.4 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
763expa 1115 . . . . 5 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑧 = 𝐶) → (𝜑𝜓))
87pm5.74da 803 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑧 = 𝐶𝜑) ↔ (𝑧 = 𝐶𝜓)))
94, 5, 8, 3vtocl2 3536 . . 3 (𝑧 = 𝐶𝜓)
103, 92thd 268 . 2 (𝑧 = 𝐶 → (𝜑𝜓))
111, 10, 2vtocl 3534 1 𝜓
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2114  Vcvv 3469 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-cleq 2815  df-clel 2894 This theorem is referenced by: (None)
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