ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  vtoclg1f GIF version

Theorem vtoclg1f 2796
Description: Version of vtoclgf 2795 with one nonfreeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 1506 and ax-13 2150. (Contributed by BJ, 1-May-2019.)
Hypotheses
Ref Expression
vtoclg1f.nf 𝑥𝜓
vtoclg1f.maj (𝑥 = 𝐴 → (𝜑𝜓))
vtoclg1f.min 𝜑
Assertion
Ref Expression
vtoclg1f (𝐴𝑉𝜓)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem vtoclg1f
StepHypRef Expression
1 elex 2748 . 2 (𝐴𝑉𝐴 ∈ V)
2 isset 2743 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3 vtoclg1f.nf . . . 4 𝑥𝜓
4 vtoclg1f.min . . . . 5 𝜑
5 vtoclg1f.maj . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
64, 5mpbii 148 . . . 4 (𝑥 = 𝐴𝜓)
73, 6exlimi 1594 . . 3 (∃𝑥 𝑥 = 𝐴𝜓)
82, 7sylbi 121 . 2 (𝐴 ∈ V → 𝜓)
91, 8syl 14 1 (𝐴𝑉𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wnf 1460  wex 1492  wcel 2148  Vcvv 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739
This theorem is referenced by:  opeliunxp2f  6236  summodclem2a  11382  fprodsplit1f  11635
  Copyright terms: Public domain W3C validator