dfbi3dc - Intuitionistic Logic Explorer

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Theorem dfbi3dc 1397

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 842	. . . 4 DECID DECID
2		xordc 1392	. . . . 5 DECID DECID $/\$ $\/$ $/\$
3	2	imp 124	. . . 4 DECID $/\$ DECID $/\$ $\/$ $/\$
4	1, 3	sylan2 286	. . 3 DECID $/\$ DECID $/\$ $\/$ $/\$
5		pm5.18dc 883	. . . 4 DECID DECID
6	5	imp 124	. . 3 DECID $/\$ DECID
7		notnotbdc 872	. . . . . 6 DECID
8	7	anbi2d 464	. . . . 5 DECID $/\$ $/\$
9		ancom 266	. . . . . 6 $/\$ $/\$
10	9	a1i 9	. . . . 5 DECID $/\$ $/\$
11	8, 10	orbi12d 793	. . . 4 DECID $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
12	11	adantl 277	. . 3 DECID $/\$ DECID $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
13	4, 6, 12	3bitr4d 220	. 2 DECID $/\$ DECID $/\$ $\/$ $/\$
14	13	ex 115	1 DECID DECID $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 708 DECID wdc 834

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-xor 1376

This theorem is referenced by: pm5.24dc 1398

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