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Mirrors > Home > ILE Home > Th. List > ax-i12 | GIF version |
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 1454 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Ref | Expression |
---|---|
ax-i12 | ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vz | . . . 4 setvar 𝑧 | |
2 | vx | . . . 4 setvar 𝑥 | |
3 | 1, 2 | weq 1444 | . . 3 wff 𝑧 = 𝑥 |
4 | 3, 1 | wal 1294 | . 2 wff ∀𝑧 𝑧 = 𝑥 |
5 | vy | . . . . 5 setvar 𝑦 | |
6 | 1, 5 | weq 1444 | . . . 4 wff 𝑧 = 𝑦 |
7 | 6, 1 | wal 1294 | . . 3 wff ∀𝑧 𝑧 = 𝑦 |
8 | 2, 5 | weq 1444 | . . . . 5 wff 𝑥 = 𝑦 |
9 | 8, 1 | wal 1294 | . . . . 5 wff ∀𝑧 𝑥 = 𝑦 |
10 | 8, 9 | wi 4 | . . . 4 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) |
11 | 10, 1 | wal 1294 | . . 3 wff ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) |
12 | 7, 11 | wo 667 | . 2 wff (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) |
13 | 4, 12 | wo 667 | 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 1454 ax12or 1455 dveeq2 1750 dveeq2or 1751 dvelimALT 1941 dvelimfv 1942 |
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