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

Theorem sbcth 2875
Description: A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.)
Hypothesis
Ref Expression
sbcth.1 𝜑
Assertion
Ref Expression
sbcth (𝐴𝑉[𝐴 / 𝑥]𝜑)

Proof of Theorem sbcth
StepHypRef Expression
1 sbcth.1 . . 3 𝜑
21ax-gen 1393 . 2 𝑥𝜑
3 spsbc 2873 . 2 (𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
42, 3mpi 15 1 (𝐴𝑉[𝐴 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1297  wcel 1448  [wsbc 2862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-v 2643  df-sbc 2863
This theorem is referenced by:  rabrsndc  3538  iota4an  5043
  Copyright terms: Public domain W3C validator