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Mirrors > Home > MPE Home > Th. List > df-nf | Structured version Visualization version GIF version |
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 2254). An example of where this is used is
stdpc5 2193. See nf5 2270 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 2008. 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 1972. This predicate only applies to wffs. See df-nfc 2877 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 24-Sep-2016.) Convert to definition. (Revised by BJ, 6-May-2019.) |
Ref | Expression |
---|---|
df-nf | ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | 1, 2 | wnf 1777 | . 2 wff Ⅎ𝑥𝜑 |
4 | 1, 2 | wex 1773 | . . 3 wff ∃𝑥𝜑 |
5 | 1, 2 | wal 1531 | . . 3 wff ∀𝑥𝜑 |
6 | 4, 5 | wi 4 | . 2 wff (∃𝑥𝜑 → ∀𝑥𝜑) |
7 | 3, 6 | wb 205 | 1 wff (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff setvar class |
This definition is referenced by: nf2 1779 nfi 1782 nfri 1783 nfd 1784 nfrd 1785 nfbiit 1845 nfnbi 1849 nfbidv 1917 nfnf1 2143 nfbidf 2209 nf5 2270 nf6 2271 nfnf 2311 sbnf2 2346 drnf1v 2361 nfcriOLD 2885 nfcriOLDOLD 2886 bj-nfimt 36015 bj-nnfnfTEMP 36116 bj-nfnnfTEMP 36136 |
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