Theorem bdstab 13709

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

Step	Hyp	Ref	Expression
1		bdstab.1	. . . . 5 ⊢ BOUNDED 𝜑
2	1	ax-bdn 13699	. . . 4 ⊢ BOUNDED ¬ 𝜑
3	2	ax-bdn 13699	. . 3 ⊢ BOUNDED ¬ ¬ 𝜑
4	3, 1	ax-bdim 13696	. 2 ⊢ BOUNDED (¬ ¬ 𝜑 → 𝜑)
5		df-stab 821	. 2 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
6	4, 5	bd0r 13707	1 ⊢ BOUNDED STAB 𝜑

Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 820  BOUNDED wbd 13694

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-bd0 13695  ax-bdim 13696  ax-bdn 13699

This theorem depends on definitions:  df-bi 116  df-stab 821

This theorem is referenced by: (None)