Theorem dcbid 748

Description: The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)

Hypothesis

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

Assertion

Ref	Expression
dcbid	|- ( ph -> (DECID ps <-> DECID ch ) )

Proof of Theorem dcbid

Step	Hyp	Ref	Expression
1		dcbid.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	notbid 592	. . 3 |- ( ph -> ( -. ps <-> -. ch ) )
3	1, 2	orbi12d 707	. 2 |- ( ph -> ( ( ps \/ -. ps ) <-> ( ch \/ -. ch ) ) )
4		df-dc 743	. 2 |- (DECID ps <-> ( ps \/ -. ps ) )
5		df-dc 743	. 2 |- (DECID ch <-> ( ch \/ -. ch ) )
6	3, 4, 5	3bitr4g 212	1 |- ( ph -> (DECID ps <-> DECID ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   \/ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  ltdcpi  6421  enqdc  6459  enqdc1  6460  ltdcnq  6495