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

Theorem vtoclgf 3569
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 3501 . 2 (𝐴𝑉𝐴 ∈ V)
2 vtoclgf.1 . . . 4 𝑥𝐴
32issetf 3497 . . 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 2217 . . 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 1540  wex 1779  wnf 1783  wcel 2108  wnfc 2890  Vcvv 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-v 3482
This theorem is referenced by:  vtocl2gf  3572  vtocl3gf  3573  vtoclgaf  3576  elabgf  3674  fsumsplit1  15781  ssiun2sf  32572  subtr  36315  subtr2  36316  supxrgere  45344  supxrgelem  45348  supxrge  45349  fmuldfeqlem1  45597  climsuse  45623  dvnmptdivc  45953  dvmptfprodlem  45959  stoweidlem59  46074  fourierdlem31  46153  sge0fodjrnlem  46431
  Copyright terms: Public domain W3C validator