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Theorem vtocl2 3522
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 803 . . . 4 (𝑥 = 𝐴 → ((𝑦 = 𝐵𝜑) ↔ (𝑦 = 𝐵𝜓)))
74, 6, 3vtocl 3520 . . 3 (𝑦 = 𝐵𝜓)
83, 72thd 265 . 2 (𝑦 = 𝐵 → (𝜑𝜓))
91, 8, 2vtocl 3520 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3447
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 1914  ax-6 1972  ax-7 2012  ax-8 2109
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-clel 2811
This theorem is referenced by:  vtocl3  3523  caovord  7569  sornom  10221  wloglei  11695  ipodrsima  18438  mpfind  21540  mclsppslem  34241  monotoddzzfi  41313
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