nf4dc - Intuitionistic Logic Explorer

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Theorem nf4dc 1684

Description: Variable

is effectively not free in

iff

is always true or always false, given a decidability condition. The reverse direction, nf4r 1685, 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 1682	. . 3
2		imordc 898	. . 3 DECID $\/$
3	1, 2	bitrid 192	. 2 DECID $\/$
4		orcom 729	. . 3 $\/$ $\/$
5		alnex 1513	. . . 4
6	5	orbi2i 763	. . 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 709 DECID wdc 835

wal 1362

wnf 1474

wex 1506

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 710 ax-5 1461 ax-gen 1463 ax-ie2 1508 ax-4 1524 ax-ial 1548

This theorem depends on definitions: df-bi 117 df-dc 836 df-tru 1367 df-fal 1370 df-nf 1475

This theorem is referenced by: (None)

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