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Theorem dcand 938

Description: A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.) (Revised by BJ, 14-Nov-2024.)

**Hypotheses**
Ref	Expression
dcand.1	DECID
dcand.2	DECID

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

**Proof of Theorem dcand**
Step	Hyp	Ref	Expression
1		dcand.1	. . . 4 DECID
2		df-dc 840	. . . . 5 DECID $\/$
3		id 19	. . . . . . 7
4	3	intnanrd 937	. . . . . 6 $/\$
5	4	orim2i 766	. . . . 5 $\/$ $\/$ $/\$
6	2, 5	sylbi 121	. . . 4 DECID $\/$ $/\$
7	1, 6	syl 14	. . 3 $\/$ $/\$
8		dcand.2	. . . 4 DECID
9		df-dc 840	. . . . 5 DECID $\/$
10		id 19	. . . . . . 7
11	10	intnand 936	. . . . . 6 $/\$
12	11	orim2i 766	. . . . 5 $\/$ $\/$ $/\$
13	9, 12	sylbi 121	. . . 4 DECID $\/$ $/\$
14	8, 13	syl 14	. . 3 $\/$ $/\$
15		ordir 822	. . 3 $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
16	7, 14, 15	sylanbrc 417	. 2 $/\$ $\/$ $/\$
17		df-dc 840	. 2 DECID $/\$ $/\$ $\/$ $/\$
18	16, 17	sylibr 134	1 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: dcan 939 dcfi 7144 nn0n0n1ge2b 9522 fzowrddc 11174 bitsinv1 12468 gcdsupex 12473 gcdsupcl 12474 gcdaddm 12500 nnwosdc 12555 lcmval 12580 lcmcllem 12584 lcmledvds 12587 prmdc 12647 pclemdc 12806 infpnlem2 12878 nninfdclemcl 13014