Axiom ax-i12 1395

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.

This axiom has been modified from the original ax-12 1399 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)

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 setvar z
2		vx	. . . 4 setvar x
3	1, 2	weq 1389	. . 3 wff z = x
4	3, 1	wal 1240	. 2 wff ∀z z = x
5		vy	. . . . 5 setvar y
6	1, 5	weq 1389	. . . 4 wff z = y
7	6, 1	wal 1240	. . 3 wff ∀z z = y
8	2, 5	weq 1389	. . . . 5 wff x = y
9	8, 1	wal 1240	. . . . 5 wff ∀z x = y
10	8, 9	wi 4	. . . 4 wff (x = y → ∀z x = y)
11	10, 1	wal 1240	. . 3 wff ∀z(x = y → ∀z x = y)
12	7, 11	wo 628	. 2 wff (∀z z = y ∨ ∀z(x = y → ∀z x = y))
13	4, 12	wo 628	1 wff (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y)))

This axiom is referenced by:  ax-12  1399  ax12or  1400  dveeq2  1693  dveeq2or  1694  dvelimALT  1883  dvelimfv  1884