Definition df-sb 1829

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 2318. We can also use [𝑦 / 𝑥]𝜑 in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2236.

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 2259, sbcom2 2328 and sbid2v 2340).

Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 2134 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 2338 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 2257. When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 2313 and sb6 2312.

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 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

Detailed syntax breakdown of Definition df-sb

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4	1, 2, 3	wsb 1828	. 2 wff [𝑦 / 𝑥]𝜑
5	2, 3	weq 1822	. . . 4 wff 𝑥 = 𝑦
6	5, 1	wi 4	. . 3 wff (𝑥 = 𝑦 → 𝜑)
7	5, 1	wa 378	. . . 4 wff (𝑥 = 𝑦 ∧ 𝜑)
8	7, 2	wex 1692	. . 3 wff ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)
9	6, 8	wa 378	. 2 wff ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
10	4, 9	wb 191	1 wff ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

This definition is referenced by:  sbequ2  1830  sb1  1831  sbequ8  1833  sbimi  1834  sbequ1  2130  sb2  2235  drsb1  2260  sbn  2274  subsym1  31236  bj-sb2v  31546  bj-dfsb2  31622  frege55b  36732