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Theorem vtoclgf 3551
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 3498 . 2 (𝐴𝑉𝐴 ∈ V)
2 vtoclgf.1 . . . 4 𝑥𝐴
32issetf 3493 . . 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 236 . . . 4 (𝑥 = 𝐴𝜓)
84, 7exlimi 2219 . . 3 (∃𝑥 𝑥 = 𝐴𝜓)
93, 8sylbi 220 . 2 (𝐴 ∈ V → 𝜓)
101, 9syl 17 1 (𝐴𝑉𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wex 1781  wnf 1785  wcel 2115  wnfc 2962  Vcvv 3480
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482
This theorem is referenced by:  vtocl2gf  3556  vtocl3gf  3557  vtoclgaf  3559  elabgf  3648  ssiun2sf  30317  subtr  33689  subtr2  33690  supxrgere  41831  supxrgelem  41835  supxrge  41836  fsumsplit1  42080  fmuldfeqlem1  42090  fprodcnlem  42107  climsuse  42116  dvnmptdivc  42446  dvmptfprodlem  42452  stoweidlem59  42567  fourierdlem31  42646  sge0f1o  42887  sge0fodjrnlem  42921
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