Definition df-nf 1437

Description: Define the not-free predicate for wffs. This is read "𝑥 is not free in 𝜑". Not-free means that the value of 𝑥 cannot affect the value of 𝜑, e.g., any occurrence of 𝑥 in 𝜑 is effectively bound by a "for all" or something that expands to one (such as "there exists"). In particular, substitution for a variable not free in a wff does not affect its value (sbf 1750). An example of where this is used is stdpc5 1563. See nf2 1646 for an alternate definition which does not involve nested quantifiers on the same variable.

Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition.

To be precise, our definition really means "effectively not free," because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example, 𝑥 is effectively not free in the bare expression 𝑥 = 𝑥, even though 𝑥 would be considered free in the usual textbook definition, because the value of 𝑥 in the expression 𝑥 = 𝑥 cannot affect the truth of the expression (and thus substitution will not change the result). (Contributed by Mario Carneiro, 11-Aug-2016.)

Assertion

Ref	Expression
df-nf	⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Detailed syntax breakdown of Definition df-nf

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3	1, 2	wnf 1436	. 2 wff Ⅎ𝑥𝜑
4	1, 2	wal 1329	. . . 4 wff ∀𝑥𝜑
5	1, 4	wi 4	. . 3 wff (𝜑 → ∀𝑥𝜑)
6	5, 2	wal 1329	. 2 wff ∀𝑥(𝜑 → ∀𝑥𝜑)
7	3, 6	wb 104	1 wff (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

This definition is referenced by:  nfi  1438  nfbii  1449  nfr  1498  nfd  1503  nfbidf  1519  nfnf1  1523  nford  1546  nfand  1547  nfnf  1556  nfalt  1557  19.21t  1561  nfimd  1564  19.9t  1621  nfnt  1634  nf2  1646  drnf1  1711  drnf2  1712  nfexd  1734  dveeq2or  1788  nfsb2or  1809  nfdv  1849  nfsbxy  1915  nfsbxyt  1916  sbcomxyyz  1945  sbnf2  1956  dvelimALT  1985  dvelimfv  1986  nfsb4t  1989  dvelimor  1993  oprabidlem  5802  bj-nfalt  12971