Definition df-nf 1461

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 1777). An example of where this is used is stdpc5 1584. See nf2 1668 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 1702. (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 1460	. 2 wff Ⅎ𝑥𝜑
4	1, 2	wal 1351	. . . 4 wff ∀𝑥𝜑
5	1, 4	wi 4	. . 3 wff (𝜑 → ∀𝑥𝜑)
6	5, 2	wal 1351	. 2 wff ∀𝑥(𝜑 → ∀𝑥𝜑)
7	3, 6	wb 105	1 wff (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

This definition is referenced by:  nfi  1462  nfbii  1473  nfr  1518  nfd  1523  nfbidf  1539  nfnf1  1544  nford  1567  nfand  1568  nfal  1576  nfnf  1577  nfalt  1578  19.21t  1582  nfimd  1585  19.9t  1642  nfnt  1656  nf2  1668  drnf1  1733  drnf2  1734  nfexd  1761  dveeq2or  1816  nfsb2or  1837  nfdv  1877  nfsbxy  1942  nfsbxyt  1943  sbcomxyyz  1972  sbnf2  1981  dvelimALT  2010  dvelimfv  2011  nfsb4t  2014  dvelimor  2018  oprabidlem  5905  bj-nfalt  14486