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Definition df-sb 1879
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 2433. We can also use [𝑦 / 𝑥]𝜑 in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2351.

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 2374, sbcom2 2443 and sbid2v 2455).

Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 2112 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 2453 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 2372. When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 2428 and sb6 2427.

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 1878 . 2 wff [𝑦 / 𝑥]𝜑
52, 3weq 1872 . . . 4 wff 𝑥 = 𝑦
65, 1wi 4 . . 3 wff (𝑥 = 𝑦𝜑)
75, 1wa 384 . . . 4 wff (𝑥 = 𝑦𝜑)
87, 2wex 1702 . . 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  1880  sb1  1881  sbequ8  1883  sbimi  1884  sbequ1  2108  sb2  2350  drsb1  2375  sbn  2389  subsym1  32401  bj-sb2v  32728  bj-dfsb2  32800  frege55b  38011
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