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Definition df-nf 1697
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 2263). An example of where this is used is stdpc5 2043. See nf2 2093 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 1889), 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 2635 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 1696 . 2 wff 𝑥𝜑
41, 2wal 1466 . . . 4 wff 𝑥𝜑
51, 4wi 4 . . 3 wff (𝜑 → ∀𝑥𝜑)
65, 2wal 1466 . 2 wff 𝑥(𝜑 → ∀𝑥𝜑)
73, 6wb 191 1 wff (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
This definition is referenced by:  nfi  1703  nfbii  1726  nfdv  1814  nfr  2004  nfd  2009  nfbidf  2018  19.9d  2021  19.9tOLD  2025  nfnf1  2034  nfnt  2035  nfimd  2053  nfnf  2084  nf2  2093  drnf1  2212  axie2  2480  xfree  28260  bj-nfdt0  31473  bj-nfalt  31490  bj-nfext  31491  bj-nfs1t  31502  bj-drnf1v  31541  bj-sbnf  31625  wl-sbnf1  32114  hbexg  37279
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