nfnbi - Metamath Proof Explorer

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Theorem nfnbi 1781

Description: A variable is non-free in a proposition if and only if it is so in its negation. (Contributed by BJ, 6-May-2019.)

**Assertion**
Ref	Expression
nfnbi

**Proof of Theorem nfnbi**
Step	Hyp	Ref	Expression
1		notnotb 304	. . . . 5
2	1	albii 1747	. . . 4
3	2	orbi1i 542	. . 3 $\/$ $\/$
4		orcom 402	. . 3 $\/$ $\/$
5	3, 4	bitri 264	. 2 $\/$ $\/$
6		nf3 1712	. 2 $\/$
7		nf3 1712	. 2 $\/$
8	5, 6, 7	3bitr4i 292	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wb 196 $\/$ wo 383

wal 1481

wnf 1708

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737

This theorem depends on definitions: df-bi 197 df-or 385 df-ex 1705 df-nf 1710

This theorem is referenced by: nfnt 1782