Definition df-nf 1792

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 2284). An example of where this is used is stdpc5 2222. See nf5 2295 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 2021.

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 1984.

This predicate only applies to wffs. See df-nfc 2890 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 1791	. 2 wff Ⅎ𝑥𝜑
4	1, 2	wex 1787	. . 3 wff ∃𝑥𝜑
5	1, 2	wal 1546	. . 3 wff ∀𝑥𝜑
6	4, 5	wi 4	. 2 wff (∃𝑥𝜑 → ∀𝑥𝜑)
7	3, 6	wb 208	1 wff (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

This definition is referenced by:  nf2  1793  nfi  1796  nfri  1797  nfd  1798  nfrd  1799  nfbiit  1859  nfnbi  1863  nfbidv  1930  nfnf1  2167  nfbidf  2238  nf5  2295  nf6  2296  sbnf  2324  nfnf  2337  sbnf2  2368  drnf1v  2381  bj-nfimt  36976  bj-nnfnfTEMP  37096  bj-nfnnfTEMP  37138