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Axiom ax-i12 1507
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 1512 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias ax12or 1508 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
StepHypRef Expression
1 vz . . . 4 setvar 𝑧
2 vx . . . 4 setvar 𝑥
31, 2weq 1503 . . 3 wff 𝑧 = 𝑥
43, 1wal 1351 . 2 wff 𝑧 𝑧 = 𝑥
5 vy . . . . 5 setvar 𝑦
61, 5weq 1503 . . . 4 wff 𝑧 = 𝑦
76, 1wal 1351 . . 3 wff 𝑧 𝑧 = 𝑦
82, 5weq 1503 . . . . 5 wff 𝑥 = 𝑦
98, 1wal 1351 . . . . 5 wff 𝑧 𝑥 = 𝑦
108, 9wi 4 . . . 4 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
1110, 1wal 1351 . . 3 wff 𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
127, 11wo 708 . 2 wff (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
134, 12wo 708 1 wff (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Colors of variables: wff set class
This axiom is referenced by:  ax12or  1508
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