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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 1489 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 1479 | . . 3 wff 𝑧 = 𝑥 |
4 | 3, 1 | wal 1329 | . 2 wff ∀𝑧 𝑧 = 𝑥 |
5 | vy | . . . . 5 setvar 𝑦 | |
6 | 1, 5 | weq 1479 | . . . 4 wff 𝑧 = 𝑦 |
7 | 6, 1 | wal 1329 | . . 3 wff ∀𝑧 𝑧 = 𝑦 |
8 | 2, 5 | weq 1479 | . . . . 5 wff 𝑥 = 𝑦 |
9 | 8, 1 | wal 1329 | . . . . 5 wff ∀𝑧 𝑥 = 𝑦 |
10 | 8, 9 | wi 4 | . . . 4 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) |
11 | 10, 1 | wal 1329 | . . 3 wff ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) |
12 | 7, 11 | wo 697 | . 2 wff (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) |
13 | 4, 12 | wo 697 | 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 1489 ax12or 1490 dveeq2 1787 dveeq2or 1788 dvelimALT 1985 dvelimfv 1986 |
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