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Theorem vtoclgf 3555
Description: Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of disjoint variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
vtoclgf.1 𝑥𝐴
vtoclgf.2 𝑥𝜓
vtoclgf.3 (𝑥 = 𝐴 → (𝜑𝜓))
vtoclgf.4 𝜑
Assertion
Ref Expression
vtoclgf (𝐴𝑉𝜓)

Proof of Theorem vtoclgf
StepHypRef Expression
1 elex 3490 . 2 (𝐴𝑉𝐴 ∈ V)
2 vtoclgf.1 . . . 4 𝑥𝐴
32issetf 3486 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 vtoclgf.2 . . . 4 𝑥𝜓
5 vtoclgf.4 . . . . 5 𝜑
6 vtoclgf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6mpbii 232 . . . 4 (𝑥 = 𝐴𝜓)
84, 7exlimi 2206 . . 3 (∃𝑥 𝑥 = 𝐴𝜓)
93, 8sylbi 216 . 2 (𝐴 ∈ V → 𝜓)
101, 9syl 17 1 (𝐴𝑉𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wex 1774  wnf 1778  wcel 2099  wnfc 2879  Vcvv 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-v 3473
This theorem is referenced by:  vtocl2gf  3558  vtocl3gf  3559  vtoclgaf  3562  elabgf  3663  fsumsplit1  15724  ssiun2sf  32363  subtr  35798  subtr2  35799  supxrgere  44715  supxrgelem  44719  supxrge  44720  fmuldfeqlem1  44970  climsuse  44996  dvnmptdivc  45326  dvmptfprodlem  45332  stoweidlem59  45447  fourierdlem31  45526  sge0f1o  45770  sge0fodjrnlem  45804
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