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Definition df-nf 1391
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 1701). An example of where this is used is stdpc5 1517. See nf2 1599 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, 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). (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 1390 . 2 wff 𝑥𝜑
41, 2wal 1283 . . . 4 wff 𝑥𝜑
51, 4wi 4 . . 3 wff (𝜑 → ∀𝑥𝜑)
65, 2wal 1283 . 2 wff 𝑥(𝜑 → ∀𝑥𝜑)
73, 6wb 103 1 wff (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
Colors of variables: wff set class
This definition is referenced by:  nfi  1392  nfbii  1403  nfr  1452  nfd  1457  nfbidf  1473  nfnf1  1477  nford  1500  nfand  1501  nfnf  1510  nfalt  1511  19.21t  1515  nfimd  1518  19.9t  1574  nfnt  1587  nf2  1599  drnf1  1662  drnf2  1663  nfexd  1685  dveeq2or  1738  nfsb2or  1759  nfdv  1799  nfsbxy  1860  nfsbxyt  1861  sbcomxyyz  1888  sbnf2  1899  dvelimALT  1928  dvelimfv  1929  nfsb4t  1932  dvelimor  1936  oprabidlem  5561  bj-nfalt  10711
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