dcim - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > dcim		Unicode version

Theorem dcim 846

Description: An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)

**Assertion**
Ref	Expression
dcim	DECID DECID DECID

**Proof of Theorem dcim**
Step	Hyp	Ref	Expression
1		df-dc 840	. 2 DECID $\/$
2		df-dc 840	. . . . . . . 8 DECID $\/$
3	2	anbi2i 457	. . . . . . 7 $/\$ DECID $/\$ $\/$
4		andi 823	. . . . . . 7 $/\$ $\/$ $/\$ $\/$ $/\$
5	3, 4	bitri 184	. . . . . 6 $/\$ DECID $/\$ $\/$ $/\$
6		pm3.4 333	. . . . . . 7 $/\$
7		annimim 690	. . . . . . 7 $/\$
8	6, 7	orim12i 764	. . . . . 6 $/\$ $\/$ $/\$ $\/$
9	5, 8	sylbi 121	. . . . 5 $/\$ DECID $\/$
10		df-dc 840	. . . . 5 DECID $\/$
11	9, 10	sylibr 134	. . . 4 $/\$ DECID DECID
12	11	ex 115	. . 3 DECID DECID
13		ax-in2 618	. . . . 5
14	13	a1d 22	. . . 4 DECID
15		orc 717	. . . . 5 $\/$
16	15, 10	sylibr 134	. . . 4 DECID
17	14, 16	syl6 33	. . 3 DECID DECID
18	12, 17	jaoi 721	. 2 $\/$ DECID DECID
19	1, 18	sylbi 121	1 DECID DECID DECID

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 713 DECID wdc 839

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 617 ax-in2 618 ax-io 714

This theorem depends on definitions: df-bi 117 df-dc 840

This theorem is referenced by: pm4.79dc 908 pm5.11dc 914 dcbi 942 annimdc 943