Definition df-sb 2069

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 2125.

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 2087, sbcom2 2163 and sbid2v 2513).

Note that our definition is valid even when 𝑥 and 𝑡 are replaced with the same variable, as sbid 2251 shows. We achieve this by applying twice Tarski's definition sb6 2089 which is valid for disjoint variables, and introducing a dummy variable 𝑦 which isolates 𝑥 from 𝑡, as in dfsb7 2279 with respect to sb5 2271. We can also achieve this by having 𝑥 free in the first conjunct and bound in the second, as the alternate definition dfsb1 2485 shows. Another version that mixes free and bound variables is dfsb3 2498. When 𝑥 and 𝑡 are distinct, we can express proper substitution with the simpler expressions of sb5 2271 and sb6 2089.

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 2485. (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 2068	. 2 wff [𝑡 / 𝑥]𝜑
5		vy	. . . . 5 setvar 𝑦
6	5, 3	weq 1967	. . . 4 wff 𝑦 = 𝑡
7	2, 5	weq 1967	. . . . . 6 wff 𝑥 = 𝑦
8	7, 1	wi 4	. . . . 5 wff (𝑥 = 𝑦 → 𝜑)
9	8, 2	wal 1537	. . . 4 wff ∀𝑥(𝑥 = 𝑦 → 𝜑)
10	6, 9	wi 4	. . 3 wff (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
11	10, 5	wal 1537	. 2 wff ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
12	4, 11	wb 205	1 wff ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

This definition is referenced by:  sbt  2070  stdpc4  2072  sbi1  2075  spsbe  2086  sbequ  2087  sb6  2089  sbal  2161  hbsbw  2171  sbequ1  2243  sbequ2  2244  sbequ2OLD  2245  dfsb7  2279  sbn  2280  nfsbvOLD  2329  sbievg  2361  sb4b  2475  sb4bOLD  2476  bj-ssbeq  34761  bj-ssbid2ALT  34771  bj-ssbid1ALT  34773