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Definition df-sb 1878
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 from the proper substitution of 𝑦 for 𝑥 in the wff 𝜑." That is, 𝑦 properly replaces 𝑥. For example, [𝑥 / 𝑦]𝑧𝑦 is the same as 𝑧𝑥, as shown in elsb4 2434. We can also use [𝑦 / 𝑥]𝜑 in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2352.

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 2375, sbcom2 2444 and sbid2v 2456).

Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 2111 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 2454 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another version that mixes free and bound variables is dfsb3 2373. When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 2429 and sb6 2428.

There are no restrictions on any of the variables, including what variables may occur in wff 𝜑. (Contributed by NM, 10-May-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 1877 . 2 wff [𝑦 / 𝑥]𝜑
52, 3weq 1871 . . . 4 wff 𝑥 = 𝑦
65, 1wi 4 . . 3 wff (𝑥 = 𝑦𝜑)
75, 1wa 384 . . . 4 wff (𝑥 = 𝑦𝜑)
87, 2wex 1701 . . 3 wff 𝑥(𝑥 = 𝑦𝜑)
96, 8wa 384 . 2 wff ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))
104, 9wb 196 1 wff ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
This definition is referenced by:  sbequ2  1879  sb1  1880  sbequ8  1882  sbimi  1883  sbequ1  2107  sb2  2351  drsb1  2376  sbn  2390  subsym1  32065  bj-sb2v  32393  bj-dfsb2  32465  frege55b  37670
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