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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 2258). An example of where this is used is
stdpc5 2197. See nf5 2272 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 2009. 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 1973. This predicate only applies to wffs. See df-nfc 2881 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 1778 | . 2 wff Ⅎ𝑥𝜑 |
4 | 1, 2 | wex 1774 | . . 3 wff ∃𝑥𝜑 |
5 | 1, 2 | wal 1532 | . . 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 1780 nfi 1783 nfri 1784 nfd 1785 nfrd 1786 nfbiit 1846 nfnbi 1850 nfbidv 1918 nfnf1 2144 nfbidf 2213 nf5 2272 nf6 2273 sbnf 2302 nfnf 2315 sbnf2 2350 drnf1v 2365 nfcriOLD 2889 nfcriOLDOLD 2890 bj-nfimt 36114 bj-nnfnfTEMP 36215 bj-nfnnfTEMP 36235 |
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