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Mathbox for Norm Megill |
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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This axiom is obsolete and should no longer be used.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 2458. (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 2043 | . . . 4 wff 𝑧 = 𝑥 |
4 | 3, 1 | wal 1629 | . . 3 wff ∀𝑧 𝑧 = 𝑥 |
5 | 4 | wn 3 | . 2 wff ¬ ∀𝑧 𝑧 = 𝑥 |
6 | vy | . . . . . 6 setvar 𝑦 | |
7 | 1, 6 | weq 2043 | . . . . 5 wff 𝑧 = 𝑦 |
8 | 7, 1 | wal 1629 | . . . 4 wff ∀𝑧 𝑧 = 𝑦 |
9 | 8 | wn 3 | . . 3 wff ¬ ∀𝑧 𝑧 = 𝑦 |
10 | 2, 6 | weq 2043 | . . . 4 wff 𝑥 = 𝑦 |
11 | 10, 1 | wal 1629 | . . . 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 34706 hbae-o 34710 ax13fromc9 34713 hbequid 34716 equid1ALT 34732 dvelimf-o 34736 ax5eq 34739 |
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