Definition df-nf 1472

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 1788). An example of where this is used is stdpc5 1595. See nf2 1679 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 1713. (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 1471	. 2 wff Ⅎ𝑥𝜑
4	1, 2	wal 1362	. . . 4 wff ∀𝑥𝜑
5	1, 4	wi 4	. . 3 wff (𝜑 → ∀𝑥𝜑)
6	5, 2	wal 1362	. 2 wff ∀𝑥(𝜑 → ∀𝑥𝜑)
7	3, 6	wb 105	1 wff (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

This definition is referenced by:  nfi  1473  nfbii  1484  nfr  1529  nfd  1534  nfbidf  1550  nfnf1  1555  nford  1578  nfand  1579  nfal  1587  nfnf  1588  nfalt  1589  19.21t  1593  nfimd  1596  19.9t  1653  nfnt  1667  nf2  1679  drnf1  1744  drnf2  1745  nfexd  1772  dveeq2or  1827  nfsb2or  1848  nfdv  1888  nfsbxy  1958  nfsbxyt  1959  sbcomxyyz  1988  sbnf2  1997  dvelimALT  2026  dvelimfv  2027  nfsb4t  2030  dvelimor  2034  oprabidlem  5949  bj-nfalt  15256