Definition df-sb 1812

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 when 𝑦 is properly substituted 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 1824.

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 1889, sbcom2 2043 and sbid2v 2052).

Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 1823 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 2047 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another alternate definition which uses a dummy variable is dfsb7a 2050.

When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 1938 and sb6 1937.

In classical logic, another possible definition is (𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑) but we do not have an intuitionistic proof that this is equivalent.

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 1811	. 2 wff [𝑦 / 𝑥]𝜑
5	2, 3	weq 1552	. . . 4 wff 𝑥 = 𝑦
6	5, 1	wi 4	. . 3 wff (𝑥 = 𝑦 → 𝜑)
7	5, 1	wa 104	. . . 4 wff (𝑥 = 𝑦 ∧ 𝜑)
8	7, 2	wex 1541	. . 3 wff ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)
9	6, 8	wa 104	. 2 wff ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
10	4, 9	wb 105	1 wff ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

This definition is referenced by:  sbimi  1813  sb1  1815  sb2  1816  sbequ1  1817  sbequ2  1818  drsb1  1848  spsbim  1892  sbequ8  1896  sbidm  1900  sb6  1937  hbsbv  1997  nfsbv  2003