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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 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.) |
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 1503 | . . 3 wff 𝑧 = 𝑥 |
4 | 3, 1 | wal 1351 | . 2 wff ∀𝑧 𝑧 = 𝑥 |
5 | vy | . . . . 5 setvar 𝑦 | |
6 | 1, 5 | weq 1503 | . . . 4 wff 𝑧 = 𝑦 |
7 | 6, 1 | wal 1351 | . . 3 wff ∀𝑧 𝑧 = 𝑦 |
8 | 2, 5 | weq 1503 | . . . . 5 wff 𝑥 = 𝑦 |
9 | 8, 1 | wal 1351 | . . . . 5 wff ∀𝑧 𝑥 = 𝑦 |
10 | 8, 9 | wi 4 | . . . 4 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) |
11 | 10, 1 | wal 1351 | . . 3 wff ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) |
12 | 7, 11 | wo 708 | . 2 wff (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) |
13 | 4, 12 | wo 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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