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Theorem dcand 934

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 836	. . . . 5 DECID $\/$
3		id 19	. . . . . . 7
4	3	intnanrd 933	. . . . . 6 $/\$
5	4	orim2i 762	. . . . 5 $\/$ $\/$ $/\$
6	2, 5	sylbi 121	. . . 4 DECID $\/$ $/\$
7	1, 6	syl 14	. . 3 $\/$ $/\$
8		dcand.2	. . . 4 DECID
9		df-dc 836	. . . . 5 DECID $\/$
10		id 19	. . . . . . 7
11	10	intnand 932	. . . . . 6 $/\$
12	11	orim2i 762	. . . . 5 $\/$ $\/$ $/\$
13	9, 12	sylbi 121	. . . 4 DECID $\/$ $/\$
14	8, 13	syl 14	. . 3 $\/$ $/\$
15		ordir 818	. . 3 $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
16	7, 14, 15	sylanbrc 417	. 2 $/\$ $\/$ $/\$
17		df-dc 836	. 2 DECID $/\$ $/\$ $\/$ $/\$
18	16, 17	sylibr 134	1 DECID $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 709 DECID wdc 835

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 615 ax-in2 616 ax-io 710

This theorem depends on definitions: df-bi 117 df-dc 836

This theorem is referenced by: dcan 935 dcfi 7047 nn0n0n1ge2b 9405 gcdsupex 12124 gcdsupcl 12125 gcdaddm 12151 nnwosdc 12206 lcmval 12231 lcmcllem 12235 lcmledvds 12238 prmdc 12298 pclemdc 12457 infpnlem2 12529 nninfdclemcl 12665