Definition df-sb 1649

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." We can also use [y / x]φ in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2024.

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 2060, sbcom2 2114 and sbid2v 2123).

Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1922 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 2119 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 2056. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 2100 and sb6 2099.

There are no restrictions on any of the variables, including what variables may occur in wff φ. (Contributed by NM, 5-Aug-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 1648	. 2 wff [y / x]φ
5	2, 3	weq 1643	. . . 4 wff x = y
6	5, 1	wi 4	. . 3 wff (x = y → φ)
7	5, 1	wa 358	. . . 4 wff (x = y ∧ φ)
8	7, 2	wex 1541	. . 3 wff ∃x(x = y ∧ φ)
9	6, 8	wa 358	. 2 wff ((x = y → φ) ∧ ∃x(x = y ∧ φ))
10	4, 9	wb 176	1 wff ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ)))

This definition is referenced by:  sbequ2  1650  sb1  1651  sbimi  1652  sbequ1  1918  drsb1  2022  sb2  2023  sbn  2062  sb6  2099