Theorem dcbi 938

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 842	. . 3 |- (DECID ph -> (DECID ps -> DECID ( ph -> ps ) ) )
2		dcim 842	. . . 4 |- (DECID ps -> (DECID ph -> DECID ( ps -> ph ) ) )
3	2	com12 30	. . 3 |- (DECID ph -> (DECID ps -> DECID ( ps -> ph ) ) )
4		dcan 935	. . . 4 |- ( (DECID ( ph -> ps ) /\ DECID ( ps -> ph ) ) -> DECID ( ( ph -> ps ) /\ ( ps -> ph ) ) )
5	4	ex 115	. . 3 |- (DECID ( ph -> ps ) -> (DECID ( ps -> ph ) -> DECID ( ( ph -> ps ) /\ ( ps -> ph ) ) ) )
6	1, 3, 5	syl6c 66	. 2 |- (DECID ph -> (DECID ps -> DECID ( ( ph -> ps ) /\ ( ps -> ph ) ) ) )
7		dfbi2 388	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
8	7	dcbii 841	. 2 |- (DECID ( ph <-> ps ) <-> DECID ( ( ph -> ps ) /\ ( ps -> ph ) ) )
9	6, 8	imbitrrdi 162	1 |- (DECID ph -> (DECID ps -> DECID ( ph <-> ps ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 835

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 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-dc 836

This theorem is referenced by:  xor3dc  1398  pm5.15dc  1400  bilukdc  1407  xordidc  1410