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Mirrors > Home > ILE Home > Th. List > ax-i12 | 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 has been modified from the original ax-12 1399 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Ref | Expression |
---|---|
ax-i12 | ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vz | . . . 4 setvar z | |
2 | vx | . . . 4 setvar x | |
3 | 1, 2 | weq 1389 | . . 3 wff z = x |
4 | 3, 1 | wal 1240 | . 2 wff ∀z z = x |
5 | vy | . . . . 5 setvar y | |
6 | 1, 5 | weq 1389 | . . . 4 wff z = y |
7 | 6, 1 | wal 1240 | . . 3 wff ∀z z = y |
8 | 2, 5 | weq 1389 | . . . . 5 wff x = y |
9 | 8, 1 | wal 1240 | . . . . 5 wff ∀z x = y |
10 | 8, 9 | wi 4 | . . . 4 wff (x = y → ∀z x = y) |
11 | 10, 1 | wal 1240 | . . 3 wff ∀z(x = y → ∀z x = y) |
12 | 7, 11 | wo 628 | . 2 wff (∀z z = y ∨ ∀z(x = y → ∀z x = y)) |
13 | 4, 12 | wo 628 | 1 wff (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))) |
Colors of variables: wff set class |
This axiom is referenced by: ax-12 1399 ax12or 1400 dveeq2 1693 dveeq2or 1694 dvelimALT 1883 dvelimfv 1884 |
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