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Definition df-nf 1700
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 2367). An example of where this is used is stdpc5 2062. See nf5 2101 for an alternate definition which involves 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 1926), 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 definition of not-free tightly ties to the quantifier 𝑥. At this state (no axioms restricting quantifiers yet) 'non-free' appears quite arbitrary. Its intended semantics expresses single-valuedness (constness) across a parameter, but is only evolved as much as later axioms assign properties to quantifiers. It seems the definition here is best suited in situations, where axioms are only partially in effect. In particular, this definition more easily carries over to other logic models with weaker axiomization.

This predicate only applies to wffs. See df-nfc 2739 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 24-Sep-2016.) Converted to definition. (Revised by BJ, 6-May-2019.)

Assertion
Ref Expression
df-nf (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀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, 2wex 1694 . . 3 wff 𝑥𝜑
51, 2wal 1472 . . 3 wff 𝑥𝜑
64, 5wi 4 . 2 wff (∃𝑥𝜑 → ∀𝑥𝜑)
73, 6wb 194 1 wff (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
This definition is referenced by:  nf2  1701  nfi  1704  nfri  1705  nfd  1706  nfrd  1707  nftht0  1708  nfbii  1769  nfnf1  2017  nf5r  2051  19.9d  2057  nfbidf  2078  nf5  2101  nf6  2102  nfnf  2143  nfeqf2  2284  sbnf2  2426  dfnf5  3905  eusv2i  4784  bj-nfdiOLD  31825
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