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Definition df-nf 1393
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 1704). An example of where this is used is stdpc5 1519. See nf2 1601 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 1392 . 2 wff 𝑥𝜑
41, 2wal 1285 . . . 4 wff 𝑥𝜑
51, 4wi 4 . . 3 wff (𝜑 → ∀𝑥𝜑)
65, 2wal 1285 . 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  1394  nfbii  1405  nfr  1454  nfd  1459  nfbidf  1475  nfnf1  1479  nford  1502  nfand  1503  nfnf  1512  nfalt  1513  19.21t  1517  nfimd  1520  19.9t  1576  nfnt  1589  nf2  1601  drnf1  1665  drnf2  1666  nfexd  1688  dveeq2or  1741  nfsb2or  1762  nfdv  1802  nfsbxy  1863  nfsbxyt  1864  sbcomxyyz  1891  sbnf2  1902  dvelimALT  1931  dvelimfv  1932  nfsb4t  1935  dvelimor  1939  oprabidlem  5637  bj-nfalt  11110
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