dcim - Intuitionistic Logic Explorer

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

Theorem dcim 784

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 743	. 2 DECID $\/$
2		df-dc 743	. . . . . . . 8 DECID $\/$
3	2	anbi2i 430	. . . . . . 7 $/\$ DECID $/\$ $\/$
4		andi 731	. . . . . . 7 $/\$ $\/$ $/\$ $\/$ $/\$
5	3, 4	bitri 173	. . . . . 6 $/\$ DECID $/\$ $\/$ $/\$
6		pm3.4 316	. . . . . . 7 $/\$
7		annimim 782	. . . . . . 7 $/\$
8	6, 7	orim12i 676	. . . . . 6 $/\$ $\/$ $/\$ $\/$
9	5, 8	sylbi 114	. . . . 5 $/\$ DECID $\/$
10		df-dc 743	. . . . 5 DECID $\/$
11	9, 10	sylibr 137	. . . 4 $/\$ DECID DECID
12	11	ex 108	. . 3 DECID DECID
13		ax-in2 545	. . . . 5
14	13	a1d 22	. . . 4 DECID
15		orc 633	. . . . 5 $\/$
16	15, 10	sylibr 137	. . . 4 DECID
17	14, 16	syl6 29	. . 3 DECID DECID
18	12, 17	jaoi 636	. 2 $\/$ DECID DECID
19	1, 18	sylbi 114	1 DECID DECID DECID

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 97 $\/$ 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: pm4.79dc 809 pm5.11dc 815 dcbi 844 annimdc 845