Definition df-nf 1510

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 1826). An example of where this is used is stdpc5 1633. 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 1509	. 2 wff Ⅎ𝑥𝜑
4	1, 2	wal 1396	. . . 4 wff ∀𝑥𝜑
5	1, 4	wi 4	. . 3 wff (𝜑 → ∀𝑥𝜑)
6	5, 2	wal 1396	. 2 wff ∀𝑥(𝜑 → ∀𝑥𝜑)
7	3, 6	wb 105	1 wff (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

This definition is referenced by:  nfi  1511  nfbii  1522  nfr  1567  nfd  1572  nfbidf  1588  nfnf1  1593  nford  1616  nfand  1617  nfal  1625  nfnf  1626  nfalt  1627  19.21t  1631  nfimd  1634  19.9t  1691  nfnt  1704  nf2  1716  drnf1  1782  drnf2  1783  nfexd  1810  dveeq2or  1865  nfsb2or  1886  nfdv  1926  nfsbxy  1996  nfsbxyt  1997  sbcomxyyz  2026  sbnf2  2035  dvelimALT  2064  dvelimfv  2065  nfsb4t  2068  dvelimor  2072  oprabidlem  6072  bj-nfalt  16506