nf4dc - Intuitionistic Logic Explorer

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Theorem nf4dc 1560

Description: Variable

is effectively not free in

iff

is always true or always false, given a decidability condition. The reverse direction, nf4r 1561, 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 1558	. . 3
2		imordc 796	. . 3 DECID $\/$
3	1, 2	syl5bb 181	. 2 DECID $\/$
4		orcom 647	. . 3 $\/$ $\/$
5		alnex 1388	. . . 4
6	5	orbi2i 679	. . 3 $\/$ $\/$
7	4, 6	bitr4i 176	. 2 $\/$ $\/$
8	3, 7	syl6bb 185	1 DECID $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wb 98 $\/$ wo 629 DECID wdc 742

wal 1241

wnf 1349

wex 1381

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-gen 1338 ax-ie2 1383 ax-4 1400 ax-ial 1427

This theorem depends on definitions: df-bi 110 df-dc 743 df-tru 1246 df-fal 1249 df-nf 1350

This theorem is referenced by: (None)

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