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Theorem vtoclgf 3568
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 3500 . 2 (𝐴𝑉𝐴 ∈ V)
2 vtoclgf.1 . . . 4 𝑥𝐴
32issetf 3496 . . 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 233 . . . 4 (𝑥 = 𝐴𝜓)
84, 7exlimi 2216 . . 3 (∃𝑥 𝑥 = 𝐴𝜓)
93, 8sylbi 217 . 2 (𝐴 ∈ V → 𝜓)
101, 9syl 17 1 (𝐴𝑉𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wex 1778  wnf 1782  wcel 2107  wnfc 2889  Vcvv 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-v 3481
This theorem is referenced by:  vtocl2gf  3571  vtocl3gf  3572  vtoclgaf  3575  elabgf  3673  fsumsplit1  15782  ssiun2sf  32573  subtr  36316  subtr2  36317  supxrgere  45349  supxrgelem  45353  supxrge  45354  fmuldfeqlem1  45602  climsuse  45628  dvnmptdivc  45958  dvmptfprodlem  45964  stoweidlem59  46079  fourierdlem31  46158  sge0fodjrnlem  46436
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