nf4dc - Intuitionistic Logic Explorer

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Theorem nf4dc 1663

Description: Variable

is effectively not free in

iff

is always true or always false, given a decidability condition. The reverse direction, nf4r 1664, 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 1661	. . 3
2		imordc 892	. . 3 DECID $\/$
3	1, 2	syl5bb 191	. 2 DECID $\/$
4		orcom 723	. . 3 $\/$ $\/$
5		alnex 1492	. . . 4
6	5	orbi2i 757	. . 3 $\/$ $\/$
7	4, 6	bitr4i 186	. 2 $\/$ $\/$
8	3, 7	bitrdi 195	1 DECID $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wb 104 $\/$ wo 703 DECID wdc 829

wal 1346

wnf 1453

wex 1485

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-gen 1442 ax-ie2 1487 ax-4 1503 ax-ial 1527

This theorem depends on definitions: df-bi 116 df-dc 830 df-tru 1351 df-fal 1354 df-nf 1454

This theorem is referenced by: (None)

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