Axiom ax-i12 1555

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 1560 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias ax12or 1556 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 1551	. . 3 wff 𝑧 = 𝑥
4	3, 1	wal 1395	. 2 wff ∀𝑧 𝑧 = 𝑥
5		vy	. . . . 5 setvar 𝑦
6	1, 5	weq 1551	. . . 4 wff 𝑧 = 𝑦
7	6, 1	wal 1395	. . 3 wff ∀𝑧 𝑧 = 𝑦
8	2, 5	weq 1551	. . . . 5 wff 𝑥 = 𝑦
9	8, 1	wal 1395	. . . . 5 wff ∀𝑧 𝑥 = 𝑦
10	8, 9	wi 4	. . . 4 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
11	10, 1	wal 1395	. . 3 wff ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
12	7, 11	wo 715	. 2 wff (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
13	4, 12	wo 715	1 wff (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

This axiom is referenced by:  ax12or  1556