Theorem bdstab 13709

Description: Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)

Hypothesis

Ref	Expression
bdstab.1	|- BOUNDED ph

Assertion

Ref	Expression
bdstab	|- BOUNDED STAB ph

Proof of Theorem bdstab

Step	Hyp	Ref	Expression
1		bdstab.1	. . . . 5 |- BOUNDED ph
2	1	ax-bdn 13699	. . . 4 |- BOUNDED -. ph
3	2	ax-bdn 13699	. . 3 |- BOUNDED -. -. ph
4	3, 1	ax-bdim 13696	. 2 |- BOUNDED ( -. -. ph -> ph )
5		df-stab 821	. 2 |- (STAB ph <-> ( -. -. ph -> ph ) )
6	4, 5	bd0r 13707	1 |- BOUNDED STAB ph

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)