nf4r - Intuitionistic Logic Explorer

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Theorem nf4r 1669

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 1668. (Contributed by Jim Kingdon, 21-Jul-2018.)

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

**Proof of Theorem nf4r**
Step	Hyp	Ref	Expression
1		orcom 728	. . 3 $\/$ $\/$
2		alnex 1497	. . . 4
3	2	orbi2i 762	. . 3 $\/$ $\/$
4	1, 3	bitr4i 187	. 2 $\/$ $\/$
5		imorr 721	. . 3 $\/$
6		nf2 1666	. . 3
7	5, 6	sylibr 134	. 2 $\/$
8	4, 7	sylbir 135	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $\/$ wo 708

wal 1351

wnf 1458

wex 1490

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 1445 ax-gen 1447 ax-ie2 1492 ax-4 1508 ax-ial 1532

This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-nf 1459

This theorem is referenced by: (None)