Theorem dcbii 747

Description: The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)

Hypothesis

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

Assertion

Ref	Expression
dcbii	|- (DECID ph <-> DECID ps )

Proof of Theorem dcbii

Step	Hyp	Ref	Expression
1		dcbii.1	. . 3 |- ( ph <-> ps )
2	1	notbii 594	. . 3 |- ( -. ph <-> -. ps )
3	1, 2	orbi12i 681	. 2 |- ( ( ph \/ -. ph ) <-> ( ps \/ -. ps ) )
4		df-dc 743	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
5		df-dc 743	. 2 |- (DECID ps <-> ( ps \/ -. ps ) )
6	3, 4, 5	3bitr4i 201	1 |- (DECID ph <-> DECID ps )

Syntax hints:   -. wn 3   <-> 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:  dcbi  844  dcned  2212  euxfr2dc  2726