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Definition df-nf 1699
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 2272). An example of where this is used is stdpc5 2039. See nf2 2090 for an alternate definition which does not involve nested quantifiers on the same variable.

Not-free is a commonly used constraint, 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 not-free notion 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 only considers syntactic freedom). For example, 𝑥 is effectively not free in the bare expression 𝑥 = 𝑥 (see nfequid 1890), even though 𝑥 would be considered free in the usual textbook definition, because the value of 𝑥 in the expression 𝑥 = 𝑥 cannot affect the truth of the expression (and thus substitution will not change the result).

This predicate only applies to wffs. See df-nfc 2644 for a not-free predicate for class variables. (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 1698 . 2 wff 𝑥𝜑
41, 2wal 1472 . . . 4 wff 𝑥𝜑
51, 4wi 4 . . 3 wff (𝜑 → ∀𝑥𝜑)
65, 2wal 1472 . 2 wff 𝑥(𝜑 → ∀𝑥𝜑)
73, 6wb 194 1 wff (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
This definition is referenced by:  nfi  1705  nfbii  1728  nfdv  1818  nfr  2004  nfd  2009  nfbidf  2017  19.9d  2020  nfnf1  2029  nfnt  2030  nfimd  2048  nfnf  2081  nf2  2090  drnf1  2221  axie2  2489  xfree  28476  bj-nfdt0  31707  bj-nfalt  31724  bj-nfext  31725  bj-nfs1t  31736  bj-drnf1v  31774  bj-sbnf  31858  wl-sbnf1  32390  hbexg  37675
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