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Definition df-sb 1688
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 1700.

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 1763, sbcom2 1906 and sbid2v 1915).

Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 1699 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 1910 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 1913.

When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 1810 and sb6 1809.

In classical logic, another possible definition is (𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑) 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.)

Assertion
Ref Expression
df-sb ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))

Detailed syntax breakdown of Definition df-sb
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
41, 2, 3wsb 1687 . 2 wff [𝑦 / 𝑥]𝜑
52, 3weq 1433 . . . 4 wff 𝑥 = 𝑦
65, 1wi 4 . . 3 wff (𝑥 = 𝑦𝜑)
75, 1wa 102 . . . 4 wff (𝑥 = 𝑦𝜑)
87, 2wex 1422 . . 3 wff 𝑥(𝑥 = 𝑦𝜑)
96, 8wa 102 . 2 wff ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))
104, 9wb 103 1 wff ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff set class
This definition is referenced by:  sbimi  1689  sb1  1691  sb2  1692  sbequ1  1693  sbequ2  1694  drsb1  1722  spsbim  1766  sbequ8  1770  sbidm  1774  sb6  1809  hbsbv  1860
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