dfbi3dc - Intuitionistic Logic Explorer

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Theorem dfbi3dc 1386

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 832	. . . 4 DECID DECID
2		xordc 1381	. . . . 5 DECID DECID $/\$ $\/$ $/\$
3	2	imp 123	. . . 4 DECID $/\$ DECID $/\$ $\/$ $/\$
4	1, 3	sylan2 284	. . 3 DECID $/\$ DECID $/\$ $\/$ $/\$
5		pm5.18dc 873	. . . 4 DECID DECID
6	5	imp 123	. . 3 DECID $/\$ DECID
7		notnotbdc 862	. . . . . 6 DECID
8	7	anbi2d 460	. . . . 5 DECID $/\$ $/\$
9		ancom 264	. . . . . 6 $/\$ $/\$
10	9	a1i 9	. . . . 5 DECID $/\$ $/\$
11	8, 10	orbi12d 783	. . . 4 DECID $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
12	11	adantl 275	. . 3 DECID $/\$ DECID $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
13	4, 6, 12	3bitr4d 219	. 2 DECID $/\$ DECID $/\$ $\/$ $/\$
14	13	ex 114	1 DECID DECID $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698 DECID wdc 824

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-stab 821 df-dc 825 df-xor 1365

This theorem is referenced by: pm5.24dc 1387

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