Axiom ax-i12 1521

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 ax12 1526 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias ax12or 1522 instead, for labeling consistency. (New usage is discouraged.)

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 1517	. . 3 wff 𝑧 = 𝑥
4	3, 1	wal 1362	. 2 wff ∀𝑧 𝑧 = 𝑥
5		vy	. . . . 5 setvar 𝑦
6	1, 5	weq 1517	. . . 4 wff 𝑧 = 𝑦
7	6, 1	wal 1362	. . 3 wff ∀𝑧 𝑧 = 𝑦
8	2, 5	weq 1517	. . . . 5 wff 𝑥 = 𝑦
9	8, 1	wal 1362	. . . . 5 wff ∀𝑧 𝑥 = 𝑦
10	8, 9	wi 4	. . . 4 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
11	10, 1	wal 1362	. . 3 wff ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
12	7, 11	wo 709	. 2 wff (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
13	4, 12	wo 709	1 wff (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

This axiom is referenced by:  ax12or  1522