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

Theorem sbcthdv 2830
Description: Deduction version of sbcth 2829. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
sbcthdv.1  |-  ( ph  ->  ps )
Assertion
Ref Expression
sbcthdv  |-  ( (
ph  /\  A  e.  V )  ->  [. A  /  x ]. ps )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    A( x)    V( x)

Proof of Theorem sbcthdv
StepHypRef Expression
1 sbcthdv.1 . . 3  |-  ( ph  ->  ps )
21alrimiv 1796 . 2  |-  ( ph  ->  A. x ps )
3 spsbc 2827 . 2  |-  ( A  e.  V  ->  ( A. x ps  ->  [. A  /  x ]. ps )
)
42, 3mpan9 275 1  |-  ( (
ph  /\  A  e.  V )  ->  [. A  /  x ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283    e. wcel 1434   [.wsbc 2816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-v 2604  df-sbc 2817
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator