pm4.79dc - Intuitionistic Logic Explorer

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Theorem pm4.79dc 898

Description: Equivalence between a disjunction of two implications, and a conjunction and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121 but with additional decidability antecedents. (Contributed by Jim Kingdon, 28-Mar-2018.)

**Assertion**
Ref	Expression
pm4.79dc	DECID DECID $\/$ $/\$

**Proof of Theorem pm4.79dc**
Step	Hyp	Ref	Expression
1		id 19	. . . 4
2		id 19	. . . 4
3	1, 2	jaoa 715	. . 3 $\/$ $/\$
4		simplimdc 855	. . . . . 6 DECID
5		pm3.3 259	. . . . . 6 $/\$
6	4, 5	syl9 72	. . . . 5 DECID $/\$
7		dcim 836	. . . . . 6 DECID DECID DECID
8		pm2.54dc 886	. . . . . 6 DECID $\/$
9	7, 8	syl6 33	. . . . 5 DECID DECID $\/$
10	6, 9	syl5d 68	. . . 4 DECID DECID $/\$ $\/$
11	10	imp 123	. . 3 DECID $/\$ DECID $/\$ $\/$
12	3, 11	impbid2 142	. 2 DECID $/\$ DECID $\/$ $/\$
13	12	expcom 115	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)

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