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

Theorem sbcthdv 2896
Description: Deduction version of sbcth 2895. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
sbcthdv.1 (𝜑𝜓)
Assertion
Ref Expression
sbcthdv ((𝜑𝐴𝑉) → [𝐴 / 𝑥]𝜓)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem sbcthdv
StepHypRef Expression
1 sbcthdv.1 . . 3 (𝜑𝜓)
21alrimiv 1830 . 2 (𝜑 → ∀𝑥𝜓)
3 spsbc 2893 . 2 (𝐴𝑉 → (∀𝑥𝜓[𝐴 / 𝑥]𝜓))
42, 3mpan9 279 1 ((𝜑𝐴𝑉) → [𝐴 / 𝑥]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1314  wcel 1465  [wsbc 2882
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-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662  df-sbc 2883
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator