dfbi3dc - Intuitionistic Logic Explorer

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Theorem dfbi3dc 1328

Description: An alternate definition of the biconditional for decidable propositions. Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability conditions. (Contributed by Jim Kingdon, 5-May-2018.)

**Assertion**
Ref	Expression
dfbi3dc	DECID DECID $/\$ $\/$ $/\$

**Proof of Theorem dfbi3dc**
Step	Hyp	Ref	Expression
1		dcn 779	. . . 4 DECID DECID
2		xordc 1323	. . . . 5 DECID DECID $/\$ $\/$ $/\$
3	2	imp 122	. . . 4 DECID $/\$ DECID $/\$ $\/$ $/\$
4	1, 3	sylan2 280	. . 3 DECID $/\$ DECID $/\$ $\/$ $/\$
5		pm5.18dc 810	. . . 4 DECID DECID
6	5	imp 122	. . 3 DECID $/\$ DECID
7		notnotbdc 799	. . . . . 6 DECID
8	7	anbi2d 451	. . . . 5 DECID $/\$ $/\$
9		ancom 262	. . . . . 6 $/\$ $/\$
10	9	a1i 9	. . . . 5 DECID $/\$ $/\$
11	8, 10	orbi12d 739	. . . 4 DECID $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
12	11	adantl 271	. . 3 DECID $/\$ DECID $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
13	4, 6, 12	3bitr4d 218	. 2 DECID $/\$ DECID $/\$ $\/$ $/\$
14	13	ex 113	1 DECID DECID $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 102

wb 103 $\/$ wo 661 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-dc 776 df-xor 1307

This theorem is referenced by: pm5.24dc 1329

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