Definition df-sb 1777

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 1789.

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 1854, sbcom2 2006 and sbid2v 2015).

Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 1788 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 2010 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 2013.

When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 1902 and sb6 1901.

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 1776	. 2 wff [𝑦 / 𝑥]𝜑
5	2, 3	weq 1517	. . . 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 1506	. . 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  1778  sb1  1780  sb2  1781  sbequ1  1782  sbequ2  1783  drsb1  1813  spsbim  1857  sbequ8  1861  sbidm  1865  sb6  1901  hbsbv  1960  nfsbv  1966