Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdstab GIF version

Theorem bdstab 10776
Description: Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
Hypothesis
Ref Expression
bdstab.1 BOUNDED 𝜑
Assertion
Ref Expression
bdstab BOUNDED STAB 𝜑

Proof of Theorem bdstab
StepHypRef Expression
1 bdstab.1 . . . . 5 BOUNDED 𝜑
21ax-bdn 10766 . . . 4 BOUNDED ¬ 𝜑
32ax-bdn 10766 . . 3 BOUNDED ¬ ¬ 𝜑
43, 1ax-bdim 10763 . 2 BOUNDED (¬ ¬ 𝜑𝜑)
5 df-stab 774 . 2 (STAB 𝜑 ↔ (¬ ¬ 𝜑𝜑))
64, 5bd0r 10774 1 BOUNDED STAB 𝜑
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  STAB wstab 773  BOUNDED wbd 10761
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-bd0 10762  ax-bdim 10763  ax-bdn 10766
This theorem depends on definitions:  df-bi 115  df-stab 774
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator