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Definition df-nf 1708
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 2378). An example of where this is used is stdpc5 2074. See nf5 2114 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 formula 𝑥 = 𝑥 (see nfequid 1938), even though 𝑥 would be considered free in the usual textbook definition, because the value of 𝑥 in the formula 𝑥 = 𝑥 cannot affect the truth of that formula (and thus substitutions 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.

The reverse implication of the definiens (the right hand side of the biconditional) always holds, see 19.2 1890.

This predicate only applies to wffs. See df-nfc 2751 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 1706 . 2 wff 𝑥𝜑
41, 2wex 1702 . . 3 wff 𝑥𝜑
51, 2wal 1479 . . 3 wff 𝑥𝜑
64, 5wi 4 . 2 wff (∃𝑥𝜑 → ∀𝑥𝜑)
73, 6wb 196 1 wff (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
This definition is referenced by:  nf2  1709  nfi  1712  nfri  1713  nfd  1714  nfrd  1715  nftht  1716  19.38a  1765  19.38b  1766  nfbiit  1775  nfimt  1819  nfnf1  2029  nf5r  2062  19.9d  2068  nfbidf  2090  nf5  2114  nf6  2115  nfnf  2156  nfeqf2  2295  sbnf2  2437  dfnf5  3943  bj-alrimhi  32579  bj-ssbft  32617
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