ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  vtoclgf GIF version

Theorem vtoclgf 2747
Description: Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of distinct 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 2700 . 2 (𝐴𝑉𝐴 ∈ V)
2 vtoclgf.1 . . . 4 𝑥𝐴
32issetf 2696 . . 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 147 . . . 4 (𝑥 = 𝐴𝜓)
84, 7exlimi 1574 . . 3 (∃𝑥 𝑥 = 𝐴𝜓)
93, 8sylbi 120 . 2 (𝐴 ∈ V → 𝜓)
101, 9syl 14 1 (𝐴𝑉𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1332  wnf 1437  wex 1469  wcel 1481  wnfc 2269  Vcvv 2689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691
This theorem is referenced by:  vtoclg  2749  vtocl2gf  2751  vtocl3gf  2752  vtoclgaf  2754  ceqsexg  2816  elabgf  2829  mob  2869  opeliunxp2  4686  fvmptss2  5503
  Copyright terms: Public domain W3C validator