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Theorem vtoclgf 3554
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 3492 . 2 (𝐴𝑉𝐴 ∈ V)
2 vtoclgf.1 . . . 4 𝑥𝐴
32issetf 3488 . . 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 2210 . . 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 1541  wex 1781  wnf 1785  wcel 2106  wnfc 2883  Vcvv 3474
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-v 3476
This theorem is referenced by:  vtocl2gf  3560  vtocl3gf  3561  vtoclgaf  3564  elabgf  3664  fsumsplit1  15695  ssiun2sf  32046  subtr  35502  subtr2  35503  supxrgere  44342  supxrgelem  44346  supxrge  44347  fmuldfeqlem1  44597  fprodcnlem  44614  climsuse  44623  dvnmptdivc  44953  dvmptfprodlem  44959  stoweidlem59  45074  fourierdlem31  45153  sge0f1o  45397  sge0fodjrnlem  45431
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