nf4dc - Intuitionistic Logic Explorer

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Theorem nf4dc 1693

Description: Variable

is effectively not free in

iff

is always true or always false, given a decidability condition. The reverse direction, nf4r 1694, 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 1691	. . 3
2		imordc 899	. . 3 DECID $\/$
3	1, 2	bitrid 192	. 2 DECID $\/$
4		orcom 730	. . 3 $\/$ $\/$
5		alnex 1522	. . . 4
6	5	orbi2i 764	. . 3 $\/$ $\/$
7	4, 6	bitr4i 187	. 2 $\/$ $\/$
8	3, 7	bitrdi 196	1 DECID $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wb 105 $\/$ wo 710 DECID wdc 836

wal 1371

wnf 1483

wex 1515

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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-gen 1472 ax-ie2 1517 ax-4 1533 ax-ial 1557

This theorem depends on definitions: df-bi 117 df-dc 837 df-tru 1376 df-fal 1379 df-nf 1484

This theorem is referenced by: (None)

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