stabtestimpdc - Intuitionistic Logic Explorer

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Theorem stabtestimpdc 857

Description: "Stable and testable" is equivalent to decidable. (Contributed by David A. Wheeler, 13-Aug-2018.)

**Assertion**
Ref	Expression
stabtestimpdc	STAB $/\$ DECID DECID

**Proof of Theorem stabtestimpdc**
Step	Hyp	Ref	Expression
1		exmiddc 777	. . . . . 6 DECID $\/$
2	1	adantl 271	. . . . 5 STAB $/\$ DECID $\/$
3		df-stab 773	. . . . . . . 8 STAB
4	3	biimpi 118	. . . . . . 7 STAB
5	4	orim2d 734	. . . . . 6 STAB $\/$ $\/$
6	5	adantr 270	. . . . 5 STAB $/\$ DECID $\/$ $\/$
7	2, 6	mpd 13	. . . 4 STAB $/\$ DECID $\/$
8	7	orcomd 680	. . 3 STAB $/\$ DECID $\/$
9		df-dc 776	. . 3 DECID $\/$
10	8, 9	sylibr 132	. 2 STAB $/\$ DECID DECID
11		dcimpstab 785	. . 3 DECID STAB
12		dcn 779	. . 3 DECID DECID
13	11, 12	jca 300	. 2 DECID STAB $/\$ DECID
14	10, 13	impbii 124	1 STAB $/\$ DECID DECID

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 102

wb 103 $\/$ wo 661 STAB wstab 772 DECID wdc 775

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 576 ax-in2 577 ax-io 662

This theorem depends on definitions: df-bi 115 df-stab 773 df-dc 776

This theorem is referenced by: (None)