Theorem stbid 832

Description: The equivalent of a stable proposition is stable. (Contributed by Jim Kingdon, 12-Aug-2022.)

Hypothesis

Ref	Expression
stbid.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
stbid	|- ( ph -> (STAB ps <-> STAB ch ) )

Proof of Theorem stbid

Step	Hyp	Ref	Expression
1		stbid.1	. . . . 5 |- ( ph -> ( ps <-> ch ) )
2	1	notbid 667	. . . 4 |- ( ph -> ( -. ps <-> -. ch ) )
3	2	notbid 667	. . 3 |- ( ph -> ( -. -. ps <-> -. -. ch ) )
4	3, 1	imbi12d 234	. 2 |- ( ph -> ( ( -. -. ps -> ps ) <-> ( -. -. ch -> ch ) ) )
5		df-stab 831	. 2 |- (STAB ps <-> ( -. -. ps -> ps ) )
6		df-stab 831	. 2 |- (STAB ch <-> ( -. -. ch -> ch ) )
7	4, 5, 6	3bitr4g 223	1 |- ( ph -> (STAB ps <-> STAB ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105  STAB wstab 830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117  df-stab 831

This theorem is referenced by:  dfss4st  3370  exmid1stab  4210  cnstab  8604