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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This axiom is obsolete and should no longer be used.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 2389. (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 1964 | . . . 4 wff 𝑧 = 𝑥 |
4 | 3, 1 | wal 1536 | . . 3 wff ∀𝑧 𝑧 = 𝑥 |
5 | 4 | wn 3 | . 2 wff ¬ ∀𝑧 𝑧 = 𝑥 |
6 | vy | . . . . . 6 setvar 𝑦 | |
7 | 1, 6 | weq 1964 | . . . . 5 wff 𝑧 = 𝑦 |
8 | 7, 1 | wal 1536 | . . . 4 wff ∀𝑧 𝑧 = 𝑦 |
9 | 8 | wn 3 | . . 3 wff ¬ ∀𝑧 𝑧 = 𝑦 |
10 | 2, 6 | weq 1964 | . . . 4 wff 𝑥 = 𝑦 |
11 | 10, 1 | wal 1536 | . . . 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 36195 hbae-o 36199 ax13fromc9 36202 hbequid 36205 equid1ALT 36221 dvelimf-o 36225 ax5eq 36228 |
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