Definition df-nf 1509

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 1825). An example of where this is used is stdpc5 1632. See nf2 1716 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 1750. (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 1508	. 2 wff Ⅎ𝑥𝜑
4	1, 2	wal 1395	. . . 4 wff ∀𝑥𝜑
5	1, 4	wi 4	. . 3 wff (𝜑 → ∀𝑥𝜑)
6	5, 2	wal 1395	. 2 wff ∀𝑥(𝜑 → ∀𝑥𝜑)
7	3, 6	wb 105	1 wff (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

This definition is referenced by:  nfi  1510  nfbii  1521  nfr  1566  nfd  1571  nfbidf  1587  nfnf1  1592  nford  1615  nfand  1616  nfal  1624  nfnf  1625  nfalt  1626  19.21t  1630  nfimd  1633  19.9t  1690  nfnt  1704  nf2  1716  drnf1  1781  drnf2  1782  nfexd  1809  dveeq2or  1864  nfsb2or  1885  nfdv  1925  nfsbxy  1995  nfsbxyt  1996  sbcomxyyz  2025  sbnf2  2034  dvelimALT  2063  dvelimfv  2064  nfsb4t  2067  dvelimor  2071  oprabidlem  6048  bj-nfalt  16360