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Theorem dcor 919

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 820	. 2 DECID $\/$
2		orc 701	. . . . . 6 $\/$
3	2	orcd 722	. . . . 5 $\/$ $\/$ $\/$
4		df-dc 820	. . . . 5 DECID $\/$ $\/$ $\/$ $\/$
5	3, 4	sylibr 133	. . . 4 DECID $\/$
6	5	a1d 22	. . 3 DECID DECID $\/$
7		df-dc 820	. . . . 5 DECID $\/$
8		olc 700	. . . . . . . . 9 $\/$
9	8	adantl 275	. . . . . . . 8 $/\$ $\/$
10	9	orcd 722	. . . . . . 7 $/\$ $\/$ $\/$ $\/$
11	10, 4	sylibr 133	. . . . . 6 $/\$ DECID $\/$
12		ioran 741	. . . . . . . . 9 $\/$ $/\$
13	12	biimpri 132	. . . . . . . 8 $/\$ $\/$
14	13	olcd 723	. . . . . . 7 $/\$ $\/$ $\/$ $\/$
15	14, 4	sylibr 133	. . . . . 6 $/\$ DECID $\/$
16	11, 15	jaodan 786	. . . . 5 $/\$ $\/$ DECID $\/$
17	7, 16	sylan2b 285	. . . 4 $/\$ DECID DECID $\/$
18	17	ex 114	. . 3 DECID DECID $\/$
19	6, 18	jaoi 705	. 2 $\/$ DECID DECID $\/$
20	1, 19	sylbi 120	1 DECID DECID DECID $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103 $\/$ wo 697 DECID wdc 819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698

This theorem depends on definitions: df-bi 116 df-dc 820

This theorem is referenced by: pm4.55dc 922 orandc 923 pm3.12dc 942 pm3.13dc 943 dn1dc 944 eueq3dc 2853 distrlem4prl 7385 distrlem4pru 7386 exfzdc 10010 lcmmndc 11732 isprm3 11788