Axiom ax-i12 1337

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 1341 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 set z
2		vx	. . . 4 set x
3	1, 2	weq 1331	. . 3 wff z = x
4	3, 1	wal 1272	. 2 wff ∀z z = x
5		vy	. . . . 5 set y
6	1, 5	weq 1331	. . . 4 wff z = y
7	6, 1	wal 1272	. . 3 wff ∀z z = y
8	2, 5	weq 1331	. . . . 5 wff x = y
9	8, 1	wal 1272	. . . . 5 wff ∀z x = y
10	8, 9	wi 4	. . . 4 wff (x = y → ∀z x = y)
11	10, 1	wal 1272	. . 3 wff ∀z(x = y → ∀z x = y)
12	7, 11	wo 609	. 2 wff (∀z z = y ∨ ∀z(x = y → ∀z x = y))
13	4, 12	wo 609	1 wff (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y)))

This axiom is referenced by:  ax-12  1341  ax12or  1342  dveeq2  1617  dveeq2or  1618  dvelimALT  1801  dvelimfv  1802