Theorem bj-dcbi 11702

Description: Equivalence property for DECID. TODO: solve conflict with dcbi 882; minimize dcbii 785 and dcbid 786 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dcbi	|- ( ( ph <-> ps ) -> (DECID ph <-> DECID ps ) )

Proof of Theorem bj-dcbi

Step	Hyp	Ref	Expression
1		id 19	. . 3 |- ( ( ph <-> ps ) -> ( ph <-> ps ) )
2		bj-notbi 11699	. . 3 |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) )
3	1, 2	orbi12d 742	. 2 |- ( ( ph <-> ps ) -> ( ( ph \/ -. ph ) <-> ( ps \/ -. ps ) ) )
4		df-dc 781	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
5		df-dc 781	. 2 |- (DECID ps <-> ( ps \/ -. ps ) )
6	3, 4, 5	3bitr4g 221	1 |- ( ( ph <-> ps ) -> (DECID ph <-> DECID ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 664  DECID wdc 780

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-in1 579  ax-in2 580  ax-io 665

This theorem depends on definitions:  df-bi 115  df-dc 781

This theorem is referenced by:  bj-d0clsepcl  11703