Definition df-nf 1454

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 1770). An example of where this is used is stdpc5 1577. See nf2 1661 for an alternate definition which does not involve nested quantifiers on the same variable.

Nonfreeness is a commonly used condition, 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 notion of nonfreeness 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 considers syntactic freedom). For example, 𝑥 is effectively not free in the expression 𝑥 = 𝑥 (even though 𝑥 is syntactically free in it, so would be considered "free" in the usual textbook definition) because the value of 𝑥 in the formula 𝑥 = 𝑥 does not affect the truth of that formula (and thus substitutions will not change the result), see nfequid 1695. (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 1453	. 2 wff Ⅎ𝑥𝜑
4	1, 2	wal 1346	. . . 4 wff ∀𝑥𝜑
5	1, 4	wi 4	. . 3 wff (𝜑 → ∀𝑥𝜑)
6	5, 2	wal 1346	. 2 wff ∀𝑥(𝜑 → ∀𝑥𝜑)
7	3, 6	wb 104	1 wff (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

This definition is referenced by:  nfi  1455  nfbii  1466  nfr  1511  nfd  1516  nfbidf  1532  nfnf1  1537  nford  1560  nfand  1561  nfal  1569  nfnf  1570  nfalt  1571  19.21t  1575  nfimd  1578  19.9t  1635  nfnt  1649  nf2  1661  drnf1  1726  drnf2  1727  nfexd  1754  dveeq2or  1809  nfsb2or  1830  nfdv  1870  nfsbxy  1935  nfsbxyt  1936  sbcomxyyz  1965  sbnf2  1974  dvelimALT  2003  dvelimfv  2004  nfsb4t  2007  dvelimor  2011  oprabidlem  5884  bj-nfalt  13799