Axiom ax-i12 1500

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 1505 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias ax12or 1501 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 1496	. . 3 wff 𝑧 = 𝑥
4	3, 1	wal 1346	. 2 wff ∀𝑧 𝑧 = 𝑥
5		vy	. . . . 5 setvar 𝑦
6	1, 5	weq 1496	. . . 4 wff 𝑧 = 𝑦
7	6, 1	wal 1346	. . 3 wff ∀𝑧 𝑧 = 𝑦
8	2, 5	weq 1496	. . . . 5 wff 𝑥 = 𝑦
9	8, 1	wal 1346	. . . . 5 wff ∀𝑧 𝑥 = 𝑦
10	8, 9	wi 4	. . . 4 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
11	10, 1	wal 1346	. . 3 wff ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
12	7, 11	wo 703	. 2 wff (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
13	4, 12	wo 703	1 wff (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

This axiom is referenced by:  ax12or  1501