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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax-c9 | Structured version Visualization version 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This axiom is obsolete and should no longer be used. It is proved above as theorem axc9 2289. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax-c9 | ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vz | . . . . 5 setvar 𝑧 | |
2 | vx | . . . . 5 setvar 𝑥 | |
3 | 1, 2 | weq 1860 | . . . 4 wff 𝑧 = 𝑥 |
4 | 3, 1 | wal 1472 | . . 3 wff ∀𝑧 𝑧 = 𝑥 |
5 | 4 | wn 3 | . 2 wff ¬ ∀𝑧 𝑧 = 𝑥 |
6 | vy | . . . . . 6 setvar 𝑦 | |
7 | 1, 6 | weq 1860 | . . . . 5 wff 𝑧 = 𝑦 |
8 | 7, 1 | wal 1472 | . . . 4 wff ∀𝑧 𝑧 = 𝑦 |
9 | 8 | wn 3 | . . 3 wff ¬ ∀𝑧 𝑧 = 𝑦 |
10 | 2, 6 | weq 1860 | . . . 4 wff 𝑥 = 𝑦 |
11 | 10, 1 | wal 1472 | . . . 4 wff ∀𝑧 𝑥 = 𝑦 |
12 | 10, 11 | wi 4 | . . 3 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) |
13 | 9, 12 | wi 4 | . 2 wff (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) |
14 | 5, 13 | wi 4 | 1 wff (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
Colors of variables: wff setvar class |
This axiom is referenced by: equid1 32998 hbae-o 33002 ax13fromc9 33005 hbequid 33008 equid1ALT 33024 dvelimf-o 33028 ax5eq 33031 |
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