nf4dc - Intuitionistic Logic Explorer

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Theorem nf4dc 1670

Description: Variable

is effectively not free in

iff

is always true or always false, given a decidability condition. The reverse direction, nf4r 1671, 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 1668	. . 3
2		imordc 897	. . 3 DECID $\/$
3	1, 2	bitrid 192	. 2 DECID $\/$
4		orcom 728	. . 3 $\/$ $\/$
5		alnex 1499	. . . 4
6	5	orbi2i 762	. . 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 708 DECID wdc 834

wal 1351

wnf 1460

wex 1492

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-gen 1449 ax-ie2 1494 ax-4 1510 ax-ial 1534

This theorem depends on definitions: df-bi 117 df-dc 835 df-tru 1356 df-fal 1359 df-nf 1461

This theorem is referenced by: (None)

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