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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax-c14 | Structured version Visualization version GIF version |
Description: Axiom of Quantifier
Introduction. One of the equality and substitution
axioms for a non-logical predicate in our predicate calculus with
equality. Axiom scheme C14' in [Megill]
p. 448 (p. 16 of the preprint).
It is redundant if we include ax-5 1990; see theorem axc14 2518. Alternately,
ax-5 1990 becomes unnecessary in principle with this
axiom, but we lose the
more powerful metalogic afforded by ax-5 1990.
We retain ax-c14 34692 here to
provide completeness for systems with the simpler metalogic that results
from omitting ax-5 1990, which might be easier to study for some
theoretical
purposes.
This axiom is obsolete and should no longer be used. It is proved above as theorem axc14 2518. (Contributed by NM, 24-Jun-1993.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax-c14 | ⊢ (¬ ∀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 2042 | . . . 4 wff 𝑧 = 𝑥 |
4 | 3, 1 | wal 1628 | . . 3 wff ∀𝑧 𝑧 = 𝑥 |
5 | 4 | wn 3 | . 2 wff ¬ ∀𝑧 𝑧 = 𝑥 |
6 | vy | . . . . . 6 setvar 𝑦 | |
7 | 1, 6 | weq 2042 | . . . . 5 wff 𝑧 = 𝑦 |
8 | 7, 1 | wal 1628 | . . . 4 wff ∀𝑧 𝑧 = 𝑦 |
9 | 8 | wn 3 | . . 3 wff ¬ ∀𝑧 𝑧 = 𝑦 |
10 | 2, 6 | wel 2145 | . . . 4 wff 𝑥 ∈ 𝑦 |
11 | 10, 1 | wal 1628 | . . . 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: ax5el 34738 ax12el 34743 |
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