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Theorem vtocl2 3510
 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 803 . . . 4 (𝑥 = 𝐴 → ((𝑦 = 𝐵𝜑) ↔ (𝑦 = 𝐵𝜓)))
74, 6, 3vtocl 3508 . . 3 (𝑦 = 𝐵𝜓)
83, 72thd 268 . 2 (𝑦 = 𝐵 → (𝜑𝜓))
91, 8, 2vtocl 3508 1 𝜓
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3442 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 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870 This theorem is referenced by:  vtocl3  3512  caovord  7350  sornom  9706  wloglei  11179  ipodrsima  17787  mpfind  20816  mclsppslem  33009  monotoddzzfi  40054
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