nf4dc - Intuitionistic Logic Explorer

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Theorem nf4dc 1600

Description: Variable

is effectively not free in

iff

is always true or always false, given a decidability condition. The reverse direction, nf4r 1601, 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 1598	. . 3
2		imordc 829	. . 3 DECID $\/$
3	1, 2	syl5bb 190	. 2 DECID $\/$
4		orcom 679	. . 3 $\/$ $\/$
5		alnex 1428	. . . 4
6	5	orbi2i 711	. . 3 $\/$ $\/$
7	4, 6	bitr4i 185	. 2 $\/$ $\/$
8	3, 7	syl6bb 194	1 DECID $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wb 103 $\/$ wo 661 DECID wdc 775

wal 1282

wnf 1389

wex 1421

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 ax-5 1376 ax-gen 1378 ax-ie2 1423 ax-4 1440 ax-ial 1467

This theorem depends on definitions: df-bi 115 df-dc 776 df-tru 1287 df-fal 1290 df-nf 1390

This theorem is referenced by: (None)

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