nf4r - Intuitionistic Logic Explorer

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Theorem nf4r 1561

Description: If

is always true or always false, then variable

is effectively not free in

. The converse holds given a decidability condition, as seen at nf4dc 1560. (Contributed by Jim Kingdon, 21-Jul-2018.)

**Assertion**
Ref	Expression
nf4r	$\/$

**Proof of Theorem nf4r**
Step	Hyp	Ref	Expression
1		orcom 647	. . 3 $\/$ $\/$
2		alnex 1388	. . . 4
3	2	orbi2i 679	. . 3 $\/$ $\/$
4	1, 3	bitr4i 176	. 2 $\/$ $\/$
5		imorr 797	. . 3 $\/$
6		nf2 1558	. . 3
7	5, 6	sylibr 137	. 2 $\/$
8	4, 7	sylbir 125	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $\/$ wo 629

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-tru 1246 df-fal 1249 df-nf 1350

This theorem is referenced by: (None)