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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This axiom is obsolete and should no longer be used.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 2301. (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 1871 | . . . 4 wff 𝑧 = 𝑥 |
4 | 3, 1 | wal 1478 | . . 3 wff ∀𝑧 𝑧 = 𝑥 |
5 | 4 | wn 3 | . 2 wff ¬ ∀𝑧 𝑧 = 𝑥 |
6 | vy | . . . . . 6 setvar 𝑦 | |
7 | 1, 6 | weq 1871 | . . . . 5 wff 𝑧 = 𝑦 |
8 | 7, 1 | wal 1478 | . . . 4 wff ∀𝑧 𝑧 = 𝑦 |
9 | 8 | wn 3 | . . 3 wff ¬ ∀𝑧 𝑧 = 𝑦 |
10 | 2, 6 | weq 1871 | . . . 4 wff 𝑥 = 𝑦 |
11 | 10, 1 | wal 1478 | . . . 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 33661 hbae-o 33665 ax13fromc9 33668 hbequid 33671 equid1ALT 33687 dvelimf-o 33691 ax5eq 33694 |
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