Theorem dcbi 877

Description: An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)

Assertion

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

Proof of Theorem dcbi

Step	Hyp	Ref	Expression
1		dcim 817	. . 3 |- (DECID ph -> (DECID ps -> DECID ( ph -> ps ) ) )
2		dcim 817	. . . 4 |- (DECID ps -> (DECID ph -> DECID ( ps -> ph ) ) )
3	2	com12 30	. . 3 |- (DECID ph -> (DECID ps -> DECID ( ps -> ph ) ) )
4		dcan 875	. . 3 |- (DECID ( ph -> ps ) -> (DECID ( ps -> ph ) -> DECID ( ( ph -> ps ) /\ ( ps -> ph ) ) ) )
5	1, 3, 4	syl6c 65	. 2 |- (DECID ph -> (DECID ps -> DECID ( ( ph -> ps ) /\ ( ps -> ph ) ) ) )
6		dfbi2 380	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
7	6	dcbii 780	. 2 |- (DECID ( ph <-> ps ) <-> DECID ( ( ph -> ps ) /\ ( ps -> ph ) ) )
8	5, 7	syl6ibr 160	1 |- (DECID ph -> (DECID ps -> DECID ( ph <-> ps ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103  DECID wdc 775

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 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  xor3dc  1318  pm5.15dc  1320  bilukdc  1327  xordidc  1330