nf4r - Intuitionistic Logic Explorer

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Theorem nf4r 1659

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 1658. (Contributed by Jim Kingdon, 21-Jul-2018.)

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

**Proof of Theorem nf4r**
Step	Hyp	Ref	Expression
1		orcom 718	. . 3 $\/$ $\/$
2		alnex 1487	. . . 4
3	2	orbi2i 752	. . 3 $\/$ $\/$
4	1, 3	bitr4i 186	. 2 $\/$ $\/$
5		imorr 711	. . 3 $\/$
6		nf2 1656	. . 3
7	5, 6	sylibr 133	. 2 $\/$
8	4, 7	sylbir 134	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $\/$ wo 698

wal 1341

wnf 1448

wex 1480

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-gen 1437 ax-ie2 1482 ax-4 1498 ax-ial 1522

This theorem depends on definitions: df-bi 116 df-tru 1346 df-fal 1349 df-nf 1449

This theorem is referenced by: (None)