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Theorem vtocl2 3504
Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
vtocl2.1 𝐴 ∈ V
vtocl2.2 𝐵 ∈ V
vtocl2.3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
vtocl2.4 𝜑
Assertion
Ref Expression
vtocl2 𝜓
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem vtocl2
StepHypRef Expression
1 vtocl2.2 . 2 𝐵 ∈ V
2 vtocl2.4 . . . 4 𝜑
32a1i 11 . . 3 (𝑦 = 𝐵𝜑)
4 vtocl2.1 . . . 4 𝐴 ∈ V
5 vtocl2.3 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
65pm5.74da 800 . . . 4 (𝑥 = 𝐴 → ((𝑦 = 𝐵𝜑) ↔ (𝑦 = 𝐵𝜓)))
74, 6, 3vtocl 3502 . . 3 (𝑦 = 𝐵𝜓)
83, 72thd 266 . 2 (𝑦 = 𝐵 → (𝜑𝜓))
91, 8, 2vtocl 3502 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  Vcvv 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762  df-cleq 2788  df-clel 2863
This theorem is referenced by:  vtocl3  3506  caovord  7215  sornom  9545  wloglei  11020  ipodrsima  17604  mpfind  20003  mclsppslem  32438  monotoddzzfi  39024
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