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Definition df-sb 1868
 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]𝑧 ∈ 𝑦 is the same as 𝑧 ∈ 𝑥, as shown in elsb4 2423. We can also use [𝑦 / 𝑥]𝜑 in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2341. 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 2364, sbcom2 2433 and sbid2v 2445). Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 2100 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 2443 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 2362. When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 2418 and sb6 2417. 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 1867 . 2 wff [𝑦 / 𝑥]𝜑
52, 3weq 1861 . . . 4 wff 𝑥 = 𝑦
65, 1wi 4 . . 3 wff (𝑥 = 𝑦𝜑)
75, 1wa 383 . . . 4 wff (𝑥 = 𝑦𝜑)
87, 2wex 1695 . . 3 wff 𝑥(𝑥 = 𝑦𝜑)
96, 8wa 383 . 2 wff ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))
104, 9wb 195 1 wff ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
 Colors of variables: wff setvar class This definition is referenced by:  sbequ2  1869  sb1  1870  sbequ8  1872  sbimi  1873  sbequ1  2096  sb2  2340  drsb1  2365  sbn  2379  subsym1  31596  bj-sb2v  31941  bj-dfsb2  32013  frege55b  37211
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