MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtocl2gf Structured version   Visualization version   GIF version

Theorem vtocl2gf 3545
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 3487 . 2 (𝐴𝑉𝐴 ∈ V)
2 vtocl2gf.3 . . 3 𝑦𝐵
3 vtocl2gf.2 . . . . 5 𝑦𝐴
43nfel1 2995 . . . 4 𝑦 𝐴 ∈ V
5 vtocl2gf.5 . . . 4 𝑦𝜒
64, 5nfim 1897 . . 3 𝑦(𝐴 ∈ V → 𝜒)
7 vtocl2gf.7 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
87imbi2d 344 . . 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 3540 . . 3 (𝐴 ∈ V → 𝜓)
142, 6, 8, 13vtoclgf 3540 . 2 (𝐵𝑊 → (𝐴 ∈ V → 𝜒))
151, 14mpan9 510 1 ((𝐴𝑉𝐵𝑊) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wnf 1785  wcel 2114  wnfc 2960  Vcvv 3469
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471
This theorem is referenced by:  vtocl3gf  3546  vtocl2gaf  3551  vtocl2dOLD  3903  offval22  7770  fmuldfeqlem1  42167
  Copyright terms: Public domain W3C validator