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Theorem vtocl2 3500
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   𝑥,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(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 801 . . . 4 (𝑥 = 𝐴 → ((𝑦 = 𝐵𝜑) ↔ (𝑦 = 𝐵𝜓)))
74, 6, 3vtocl 3498 . . 3 (𝑦 = 𝐵𝜓)
83, 72thd 264 . 2 (𝑦 = 𝐵 → (𝜑𝜓))
91, 8, 2vtocl 3498 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  Vcvv 3432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-clel 2816
This theorem is referenced by:  vtocl3  3501  caovord  7483  sornom  10033  wloglei  11507  ipodrsima  18259  mpfind  21317  mclsppslem  33545  monotoddzzfi  40764
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