Definition df-sb 2012

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 2516. We can also use [𝑦 / 𝑥]𝜑 in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2428.

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 2452, sbcom2 2523 and sbid2v 2490).

Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 2232 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 2535 shows (which some logicians may prefer because it does not mix free and bound variables). Another version that mixes free and bound variables is dfsb3 2450. When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 2251 and sb6 2250.

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 2011	. 2 wff [𝑦 / 𝑥]𝜑
5	2, 3	weq 2005	. . . 4 wff 𝑥 = 𝑦
6	5, 1	wi 4	. . 3 wff (𝑥 = 𝑦 → 𝜑)
7	5, 1	wa 386	. . . 4 wff (𝑥 = 𝑦 ∧ 𝜑)
8	7, 2	wex 1823	. . 3 wff ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)
9	6, 8	wa 386	. 2 wff ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
10	4, 9	wb 198	1 wff ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

This definition is referenced by:  sbequ2  2013  sb1  2014  sbequ8  2016  sbimi  2017  sbimdv  2018  equsb1v  2049  sbimd  2226  sbequ1  2228  sb2v  2243  sb2  2427  drsb1  2453  sbn  2467  subsym1  33013  bj-dfsb2  33404  wl-dv-sb  33927  sbtv  38124  frege55b  39157