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|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
We can also use [y /
x]φ in place of the "free for"
side condition used in traditional predicate calculus; see, for example,
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 1703, sbcom2 1844 and sbid2v 1853).
Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1639 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 1848 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 1851.
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.)
|df-sb||⊢ ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ)))|
|1||wph||. . 3 wff φ|
|2||vx||. . 3 setvar x|
|3||vy||. . 3 setvar y|
|4||1, 2, 3||wsb 1627||. 2 wff [y / x]φ|
|5||2, 3||weq 1374||. . . 4 wff x = y|
|6||5, 1||wi 4||. . 3 wff (x = y → φ)|
|7||5, 1||wa 97||. . . 4 wff (x = y ∧ φ)|
|8||7, 2||wex 1361||. . 3 wff ∃x(x = y ∧ φ)|
|9||6, 8||wa 97||. 2 wff ((x = y → φ) ∧ ∃x(x = y ∧ φ))|
|10||4, 9||wb 98||1 wff ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ)))|
|Colors of variables: wff set class|
|This definition is referenced by: sbimi 1629 sb1 1631 sb2 1632 sbequ1 1633 sbequ2 1634 drsb1 1662 spsbim 1706 sbequ8 1709 sbidm 1714 sb6 1749 hbsbv 1798|
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