nf4dc - Intuitionistic Logic Explorer

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Theorem nf4dc 1650

Description: Variable

is effectively not free in

iff

is always true or always false, given a decidability condition. The reverse direction, nf4r 1651, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)

**Assertion**
Ref	Expression
nf4dc	DECID $\/$

**Proof of Theorem nf4dc**
Step	Hyp	Ref	Expression
1		nf2 1648	. . 3
2		imordc 883	. . 3 DECID $\/$
3	1, 2	syl5bb 191	. 2 DECID $\/$
4		orcom 718	. . 3 $\/$ $\/$
5		alnex 1479	. . . 4
6	5	orbi2i 752	. . 3 $\/$ $\/$
7	4, 6	bitr4i 186	. 2 $\/$ $\/$
8	3, 7	bitrdi 195	1 DECID $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wb 104 $\/$ wo 698 DECID wdc 820

wal 1333

wnf 1440

wex 1472

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 ax-5 1427 ax-gen 1429 ax-ie2 1474 ax-4 1490 ax-ial 1514

This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1338 df-fal 1341 df-nf 1441

This theorem is referenced by: (None)

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