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Theorem vtocl2gf 3259
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)
Hypotheses
Ref Expression
vtocl2gf.1 𝑥𝐴
vtocl2gf.2 𝑦𝐴
vtocl2gf.3 𝑦𝐵
vtocl2gf.4 𝑥𝜓
vtocl2gf.5 𝑦𝜒
vtocl2gf.6 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl2gf.7 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl2gf.8 𝜑
Assertion
Ref Expression
vtocl2gf ((𝐴𝑉𝐵𝑊) → 𝜒)

Proof of Theorem vtocl2gf
StepHypRef Expression
1 elex 3203 . 2 (𝐴𝑉𝐴 ∈ V)
2 vtocl2gf.3 . . 3 𝑦𝐵
3 vtocl2gf.2 . . . . 5 𝑦𝐴
43nfel1 2781 . . . 4 𝑦 𝐴 ∈ V
5 vtocl2gf.5 . . . 4 𝑦𝜒
64, 5nfim 1827 . . 3 𝑦(𝐴 ∈ V → 𝜒)
7 vtocl2gf.7 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
87imbi2d 330 . . 3 (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒)))
9 vtocl2gf.1 . . . 4 𝑥𝐴
10 vtocl2gf.4 . . . 4 𝑥𝜓
11 vtocl2gf.6 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
12 vtocl2gf.8 . . . 4 𝜑
139, 10, 11, 12vtoclgf 3255 . . 3 (𝐴 ∈ V → 𝜓)
142, 6, 8, 13vtoclgf 3255 . 2 (𝐵𝑊 → (𝐴 ∈ V → 𝜒))
151, 14mpan9 486 1 ((𝐴𝑉𝐵𝑊) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wnf 1705  wcel 1992  wnfc 2754  Vcvv 3191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193
This theorem is referenced by:  vtocl3gf  3260  vtocl2g  3261  vtocl2gaf  3264  offval22  7199  vtocl2d  29155  fmuldfeqlem1  39205
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