pm4.55dc - Intuitionistic Logic Explorer

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Theorem pm4.55dc 933

Description: Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.)

**Assertion**
Ref	Expression
pm4.55dc	DECID DECID $/\$ $\/$

**Proof of Theorem pm4.55dc**
Step	Hyp	Ref	Expression
1		pm4.54dc 897	. . . . 5 DECID DECID $/\$ $\/$
2	1	imp 123	. . . 4 DECID $/\$ DECID $/\$ $\/$
3		dcn 837	. . . . . . . . 9 DECID DECID
4	3	anim2i 340	. . . . . . . 8 DECID $/\$ DECID DECID $/\$ DECID
5		dcor 930	. . . . . . . . 9 DECID DECID DECID $\/$
6	5	imp 123	. . . . . . . 8 DECID $/\$ DECID DECID $\/$
7	4, 6	syl 14	. . . . . . 7 DECID $/\$ DECID DECID $\/$
8		dcn 837	. . . . . . . . 9 DECID DECID
9		dcan2 929	. . . . . . . . 9 DECID DECID DECID $/\$
10	8, 9	syl 14	. . . . . . . 8 DECID DECID DECID $/\$
11	10	imp 123	. . . . . . 7 DECID $/\$ DECID DECID $/\$
12	7, 11	jca 304	. . . . . 6 DECID $/\$ DECID DECID $\/$ $/\$ DECID $/\$
13		con2bidc 870	. . . . . . 7 DECID $\/$ DECID $/\$ $\/$ $/\$ $/\$ $\/$
14	13	imp 123	. . . . . 6 DECID $\/$ $/\$ DECID $/\$ $\/$ $/\$ $/\$ $\/$
15	12, 14	syl 14	. . . . 5 DECID $/\$ DECID $\/$ $/\$ $/\$ $\/$
16	15	biimprd 157	. . . 4 DECID $/\$ DECID $/\$ $\/$ $\/$ $/\$
17	2, 16	mpd 13	. . 3 DECID $/\$ DECID $\/$ $/\$
18	17	bicomd 140	. 2 DECID $/\$ DECID $/\$ $\/$
19	18	ex 114	1 DECID DECID $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703 DECID wdc 829

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 609 ax-in2 610 ax-io 704

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830

This theorem is referenced by: (None)