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Definition df-nf 1448
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 1764). An example of where this is used is stdpc5 1571. See nf2 1655 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 1689. (Contributed by Mario Carneiro, 11-Aug-2016.)

Assertion
Ref Expression
df-nf (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Detailed syntax breakdown of Definition df-nf
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
31, 2wnf 1447 . 2 wff 𝑥𝜑
41, 2wal 1340 . . . 4 wff 𝑥𝜑
51, 4wi 4 . . 3 wff (𝜑 → ∀𝑥𝜑)
65, 2wal 1340 . 2 wff 𝑥(𝜑 → ∀𝑥𝜑)
73, 6wb 104 1 wff (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
Colors of variables: wff set class
This definition is referenced by:  nfi  1449  nfbii  1460  nfr  1505  nfd  1510  nfbidf  1526  nfnf1  1531  nford  1554  nfand  1555  nfal  1563  nfnf  1564  nfalt  1565  19.21t  1569  nfimd  1572  19.9t  1629  nfnt  1643  nf2  1655  drnf1  1720  drnf2  1721  nfexd  1748  dveeq2or  1803  nfsb2or  1824  nfdv  1864  nfsbxy  1929  nfsbxyt  1930  sbcomxyyz  1959  sbnf2  1968  dvelimALT  1997  dvelimfv  1998  nfsb4t  2001  dvelimor  2005  oprabidlem  5865  bj-nfalt  13507
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