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Mirrors > Home > ILE Home > Th. List > df-sb | GIF version |
Description: Define proper
substitution. Remark 9.1 in [Megill] p. 447 (p.
15 of the
preprint). For our notation, we use [𝑦 / 𝑥]𝜑 to mean "the wff
that results when 𝑦 is properly substituted for 𝑥 in the
wff
𝜑." We can also use [𝑦 / 𝑥]𝜑 in place of the "free for"
side condition used in traditional predicate calculus; see, for example,
stdpc4 1748.
Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, "𝜑(𝑦) is the wff that results when 𝑦 is properly substituted for 𝑥 in 𝜑(𝑥)." For example, if the original 𝜑(𝑥) is 𝑥 = 𝑦, then 𝜑(𝑦) is 𝑦 = 𝑦, from which we obtain that 𝜑(𝑥) is 𝑥 = 𝑥. So what exactly does 𝜑(𝑥) mean? Curry's notation solves this problem. In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 1812, sbcom2 1960 and sbid2v 1969). Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 1747 shows. We achieve this by having 𝑥 free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 1964 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another alternate definition which uses a dummy variable is dfsb7a 1967. When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 1859 and sb6 1858. In classical logic, another possible definition is (𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑) but we do not have an intuitionistic proof that this is equivalent. There are no restrictions on any of the variables, including what variables may occur in wff 𝜑. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
df-sb | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | 1, 2, 3 | wsb 1735 | . 2 wff [𝑦 / 𝑥]𝜑 |
5 | 2, 3 | weq 1479 | . . . 4 wff 𝑥 = 𝑦 |
6 | 5, 1 | wi 4 | . . 3 wff (𝑥 = 𝑦 → 𝜑) |
7 | 5, 1 | wa 103 | . . . 4 wff (𝑥 = 𝑦 ∧ 𝜑) |
8 | 7, 2 | wex 1468 | . . 3 wff ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) |
9 | 6, 8 | wa 103 | . 2 wff ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
10 | 4, 9 | wb 104 | 1 wff ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Colors of variables: wff set class |
This definition is referenced by: sbimi 1737 sb1 1739 sb2 1740 sbequ1 1741 sbequ2 1742 drsb1 1771 spsbim 1815 sbequ8 1819 sbidm 1823 sb6 1858 hbsbv 1912 |
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