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

Theorem spsbcd 2852
Description: Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1705 and rspsbc 2921. (Contributed by Mario Carneiro, 9-Feb-2017.)
Hypotheses
Ref Expression
spsbcd.1 (𝜑𝐴𝑉)
spsbcd.2 (𝜑 → ∀𝑥𝜓)
Assertion
Ref Expression
spsbcd (𝜑[𝐴 / 𝑥]𝜓)

Proof of Theorem spsbcd
StepHypRef Expression
1 spsbcd.1 . 2 (𝜑𝐴𝑉)
2 spsbcd.2 . 2 (𝜑 → ∀𝑥𝜓)
3 spsbc 2851 . 2 (𝐴𝑉 → (∀𝑥𝜓[𝐴 / 𝑥]𝜓))
41, 2, 3sylc 61 1 (𝜑[𝐴 / 𝑥]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  wcel 1438  [wsbc 2840
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-sbc 2841
This theorem is referenced by:  ovmpt2dxf  5770
  Copyright terms: Public domain W3C validator