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 2269). An example of where this is used is stdpc5 2207. See nf5 2287 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 2021), 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 1982.

This predicate only applies to wffs. See df-nfc 2902 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

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3	1, 2	wnf 1786	. 2 wff Ⅎ𝑥𝜑
4	1, 2	wex 1782	. . 3 wff ∃𝑥𝜑
5	1, 2	wal 1537	. . 3 wff ∀𝑥𝜑
6	4, 5	wi 4	. 2 wff (∃𝑥𝜑 → ∀𝑥𝜑)
7	3, 6	wb 209	1 wff (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

This definition is referenced by:  nf2  1788  nfi  1791  nfri  1792  nfd  1793  nfrd  1794  nfbiit  1853  nfbidv  1924  nfnf1  2156  nf5rOLD  2193  nfbidf  2225  nf5  2287  nf6  2288  nfnf  2335  sbnf2  2367  nfcriOLD  2910  nfcriOLDOLD  2911  bj-nfimt  34366  bj-nnfnfTEMP  34464  bj-nfnnfTEMP  34484