Axiom ax-i12 1391

Description: Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever z is distinct from x and y, and x = y is true, then x = y quantified with z is also true. In other words, z is irrelevant to the truth of x = y. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.

An open problem is whether this axiom is redundant. Note that the analogous axiom for the membership connective, ax-15 1807, has been shown to be redundant. It is also unknown whether this axiom can be replaced by a shorter formula. However, it can be derived from two slightly shorter formulas, as shown by a12study 1825.

This axiom has been modified from the original ax-12 1393 for compatibility with intuitionistic logic.

Assertion

Ref	Expression
ax-i12	⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y)))

Detailed syntax breakdown of Axiom ax-i12

Step	Hyp	Ref	Expression
1		vz	. . . 4 set z
2		vx	. . . 4 set x
3	1, 2	weq 1384	. . 3 wff z = x
4	3, 1	wal 1335	. 2 wff ∀z z = x
5		vy	. . . . 5 set y
6	1, 5	weq 1384	. . . 4 wff z = y
7	6, 1	wal 1335	. . 3 wff ∀z z = y
8	2, 5	weq 1384	. . . . 5 wff x = y
9	8, 1	wal 1335	. . . . 5 wff ∀z x = y
10	8, 9	wi 4	. . . 4 wff (x = y → ∀z x = y)
11	10, 1	wal 1335	. . 3 wff ∀z(x = y → ∀z x = y)
12	7, 11	wo 606	. 2 wff (∀z z = y ∨ ∀z(x = y → ∀z x = y))
13	4, 12	wo 606	1 wff (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y)))

This axiom is referenced by:  ax-12  1393  ax12or  1394