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Axiom ax-i12 1439
 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 1443 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
StepHypRef Expression
1 vz . . . 4 setvar 𝑧
2 vx . . . 4 setvar 𝑥
31, 2weq 1433 . . 3 wff 𝑧 = 𝑥
43, 1wal 1283 . 2 wff 𝑧 𝑧 = 𝑥
5 vy . . . . 5 setvar 𝑦
61, 5weq 1433 . . . 4 wff 𝑧 = 𝑦
76, 1wal 1283 . . 3 wff 𝑧 𝑧 = 𝑦
82, 5weq 1433 . . . . 5 wff 𝑥 = 𝑦
98, 1wal 1283 . . . . 5 wff 𝑧 𝑥 = 𝑦
108, 9wi 4 . . . 4 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
1110, 1wal 1283 . . 3 wff 𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
127, 11wo 662 . 2 wff (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
134, 12wo 662 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:  ax-12  1443  ax12or  1444  dveeq2  1738  dveeq2or  1739  dvelimALT  1929  dvelimfv  1930
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