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Definition df-nf 1787
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 2202. See nf5 2279 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 considers syntactic freedom). For example, 𝑥 is effectively not free in the formula 𝑥 = 𝑥 (even though 𝑥 is syntactically free in it, so would be considered free in the usual textbook definition) because the value of 𝑥 in the formula 𝑥 = 𝑥 does not affect the truth of that formula (and thus substitutions will not change the result), see nfequid 2017.

This definition of "not free" tightly ties to the quantifier 𝑥. At this state (no axioms restricting quantifiers yet) "nonfree" 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.

The reverse implication of the definiens (the right hand side of the biconditional) always holds, see 19.2 1981.

This predicate only applies to wffs. See df-nfc 2886 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 24-Sep-2016.) Convert 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 1786 . 2 wff 𝑥𝜑
41, 2wex 1782 . . 3 wff 𝑥𝜑
51, 2wal 1540 . . 3 wff 𝑥𝜑
64, 5wi 4 . 2 wff (∃𝑥𝜑 → ∀𝑥𝜑)
73, 6wb 205 1 wff (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
This definition is referenced by:  nf2  1788  nfi  1791  nfri  1792  nfd  1793  nfrd  1794  nfbiit  1854  nfnbi  1858  nfbidv  1926  nfnf1  2152  nfbidf  2218  nf5  2279  nf6  2280  nfnf  2320  sbnf2  2355  drnf1v  2370  nfcriOLD  2894  nfcriOLDOLD  2895  bj-nfimt  35453  bj-nnfnfTEMP  35554  bj-nfnnfTEMP  35574
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