dcor - Intuitionistic Logic Explorer

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

Theorem dcor 877

Description: A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)

**Assertion**
Ref	Expression
dcor	DECID DECID DECID $\/$

**Proof of Theorem dcor**
Step	Hyp	Ref	Expression
1		df-dc 777	. 2 DECID $\/$
2		orc 666	. . . . . 6 $\/$
3	2	orcd 685	. . . . 5 $\/$ $\/$ $\/$
4		df-dc 777	. . . . 5 DECID $\/$ $\/$ $\/$ $\/$
5	3, 4	sylibr 132	. . . 4 DECID $\/$
6	5	a1d 22	. . 3 DECID DECID $\/$
7		df-dc 777	. . . . 5 DECID $\/$
8		olc 665	. . . . . . . . 9 $\/$
9	8	adantl 271	. . . . . . . 8 $/\$ $\/$
10	9	orcd 685	. . . . . . 7 $/\$ $\/$ $\/$ $\/$
11	10, 4	sylibr 132	. . . . . 6 $/\$ DECID $\/$
12		ioran 702	. . . . . . . . 9 $\/$ $/\$
13	12	biimpri 131	. . . . . . . 8 $/\$ $\/$
14	13	olcd 686	. . . . . . 7 $/\$ $\/$ $\/$ $\/$
15	14, 4	sylibr 132	. . . . . 6 $/\$ DECID $\/$
16	11, 15	jaodan 744	. . . . 5 $/\$ $\/$ DECID $\/$
17	7, 16	sylan2b 281	. . . 4 $/\$ DECID DECID $\/$
18	17	ex 113	. . 3 DECID DECID $\/$
19	6, 18	jaoi 669	. 2 $\/$ DECID DECID $\/$
20	1, 19	sylbi 119	1 DECID DECID DECID $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 102 $\/$ wo 662 DECID wdc 776

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 577 ax-in2 578 ax-io 663

This theorem depends on definitions: df-bi 115 df-dc 777

This theorem is referenced by: pm4.55dc 880 orandc 881 pm3.12dc 900 pm3.13dc 901 dn1dc 902 eueq3dc 2767 distrlem4prl 6825 distrlem4pru 6826 exfzdc 9315 lcmmndc 10577 isprm3 10633