Definition df-nf 1485

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 1801). An example of where this is used is stdpc5 1608. See nf2 1692 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 1726. (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 1484	. 2 wff Ⅎ𝑥𝜑
4	1, 2	wal 1371	. . . 4 wff ∀𝑥𝜑
5	1, 4	wi 4	. . 3 wff (𝜑 → ∀𝑥𝜑)
6	5, 2	wal 1371	. 2 wff ∀𝑥(𝜑 → ∀𝑥𝜑)
7	3, 6	wb 105	1 wff (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

This definition is referenced by:  nfi  1486  nfbii  1497  nfr  1542  nfd  1547  nfbidf  1563  nfnf1  1568  nford  1591  nfand  1592  nfal  1600  nfnf  1601  nfalt  1602  19.21t  1606  nfimd  1609  19.9t  1666  nfnt  1680  nf2  1692  drnf1  1757  drnf2  1758  nfexd  1785  dveeq2or  1840  nfsb2or  1861  nfdv  1901  nfsbxy  1971  nfsbxyt  1972  sbcomxyyz  2001  sbnf2  2010  dvelimALT  2039  dvelimfv  2040  nfsb4t  2043  dvelimor  2047  oprabidlem  5993  bj-nfalt  15870