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Axiom ax-i12 1470
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 1474 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 1464 . . 3 wff 𝑧 = 𝑥
43, 1wal 1314 . 2 wff 𝑧 𝑧 = 𝑥
5 vy . . . . 5 setvar 𝑦
61, 5weq 1464 . . . 4 wff 𝑧 = 𝑦
76, 1wal 1314 . . 3 wff 𝑧 𝑧 = 𝑦
82, 5weq 1464 . . . . 5 wff 𝑥 = 𝑦
98, 1wal 1314 . . . . 5 wff 𝑧 𝑥 = 𝑦
108, 9wi 4 . . . 4 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
1110, 1wal 1314 . . 3 wff 𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
127, 11wo 682 . 2 wff (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
134, 12wo 682 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  1474  ax12or  1475  dveeq2  1771  dveeq2or  1772  dvelimALT  1963  dvelimfv  1964
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