Definition df-sb 2062

Description: Define proper substitution. For our notation, we use [𝑡 / 𝑥]𝜑 to mean "the wff that results from the proper substitution of 𝑡 for 𝑥 in the wff 𝜑". That is, 𝑡 properly replaces 𝑥. For example, [𝑡 / 𝑥]𝑧 ∈ 𝑥 is the same as 𝑧 ∈ 𝑡 (when 𝑥 and 𝑧 are distinct), as shown in elsb2 2122.

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(𝑡) is the wff that results when 𝑡 is properly substituted for 𝑥 in 𝜑(𝑥)". For example, if the original 𝜑(𝑥) is 𝑥 = 𝑡, then 𝜑(𝑡) is 𝑡 = 𝑡, from which we obtain that 𝜑(𝑥) is 𝑥 = 𝑥. So what exactly does 𝜑(𝑥) mean? Curry's notation solves this problem.

A very similar notation, namely (𝑦 ∣ 𝑥)𝜑, was introduced in Bourbaki's Set Theory (Chapter 1, Description of Formal Mathematic, 1953).

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 2080, sbcom2 2170 and sbid2v 2511).

Note that our definition is valid even when 𝑥 and 𝑡 are replaced with the same variable, as sbid 2252 shows. We achieve this by applying twice Tarski's definition sb6 2082 which is valid for disjoint variables, and introducing a dummy variable 𝑦 which isolates 𝑥 from 𝑡, as in dfsb7 2277 with respect to sb5 2273. We can also achieve this by having 𝑥 free in the first conjunct and bound in the second, as the alternate definition dfsb1 2483 shows. Another version that mixes free and bound variables is dfsb3 2496. When 𝑥 and 𝑡 are distinct, we can express proper substitution with the simpler expressions of sb5 2273 and sb6 2082.

Note that the occurrences of a given variable in the definiens are either all bound (𝑥, 𝑦) or all free (𝑡). Also note that the definiens uses only primitive symbols.

This double level definition will make several proofs using it appear as doubled. Alternately, one could often first prove as a lemma the same theorem with a disjoint variable condition on the substitute and the substituted variables, and then prove the original theorem by applying this lemma twice in a row. (Contributed by NM, 10-May-1993.) Revised from the original definition dfsb1 2483. (Revised by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
df-sb	⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Distinct variable groups:   𝑥,𝑦   𝑦,𝑡   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑡)

Detailed syntax breakdown of Definition df-sb

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3		vt	. . 3 setvar 𝑡
4	1, 2, 3	wsb 2061	. 2 wff [𝑡 / 𝑥]𝜑
5		vy	. . . . 5 setvar 𝑦
6	5, 3	weq 1959	. . . 4 wff 𝑦 = 𝑡
7	2, 5	weq 1959	. . . . . 6 wff 𝑥 = 𝑦
8	7, 1	wi 4	. . . . 5 wff (𝑥 = 𝑦 → 𝜑)
9	8, 2	wal 1534	. . . 4 wff ∀𝑥(𝑥 = 𝑦 → 𝜑)
10	6, 9	wi 4	. . 3 wff (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
11	10, 5	wal 1534	. 2 wff ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
12	4, 11	wb 206	1 wff ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

This definition is referenced by:  sbt  2063  stdpc4  2065  sbi1  2068  spsbe  2079  sbequ  2080  sb6  2082  sbal  2166  hbsbwOLD  2169  sbequ1  2245  sbequ2  2246  dfsb7  2277  sbn  2278  sbrim  2302  nfsbvOLD  2329  cbvsbvf  2363  sb4b  2477  sbequbidv  36196  cbvsbdavw  36236  cbvsbdavw2  36237  bj-ssbeq  36635  bj-ssbid2ALT  36645  bj-ssbid1ALT  36647