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Theorem dcim 827

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 821	. 2 DECID $\/$
2		df-dc 821	. . . . . . . 8 DECID $\/$
3	2	anbi2i 453	. . . . . . 7 $/\$ DECID $/\$ $\/$
4		andi 808	. . . . . . 7 $/\$ $\/$ $/\$ $\/$ $/\$
5	3, 4	bitri 183	. . . . . 6 $/\$ DECID $/\$ $\/$ $/\$
6		pm3.4 331	. . . . . . 7 $/\$
7		annimim 676	. . . . . . 7 $/\$
8	6, 7	orim12i 749	. . . . . 6 $/\$ $\/$ $/\$ $\/$
9	5, 8	sylbi 120	. . . . 5 $/\$ DECID $\/$
10		df-dc 821	. . . . 5 DECID $\/$
11	9, 10	sylibr 133	. . . 4 $/\$ DECID DECID
12	11	ex 114	. . 3 DECID DECID
13		ax-in2 605	. . . . 5
14	13	a1d 22	. . . 4 DECID
15		orc 702	. . . . 5 $\/$
16	15, 10	sylibr 133	. . . 4 DECID
17	14, 16	syl6 33	. . 3 DECID DECID
18	12, 17	jaoi 706	. 2 $\/$ DECID DECID
19	1, 18	sylbi 120	1 DECID DECID DECID

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103 $\/$ wo 698 DECID wdc 820

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 604 ax-in2 605 ax-io 699

This theorem depends on definitions: df-bi 116 df-dc 821

This theorem is referenced by: pm4.79dc 889 pm5.11dc 895 dcbi 921 annimdc 922