![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ax-11 | GIF version |
Description: Axiom of Variable
Substitution. One of the 5 equality axioms of predicate
calculus. The final consequent ∀x(x = y →
φ) is a way of
expressing "y substituted for x in wff φ " (cf. sb6 1763).
It
is based on Lemma 16 of [Tarski] p. 70 and
Axiom C8 of [Monk2] p. 105,
from which it can be proved by cases.
Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1705, ax11v2 1698 and ax-11o 1701. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
ax-11 | ⊢ (x = y → (∀yφ → ∀x(x = y → φ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . 3 setvar x | |
2 | vy | . . 3 setvar y | |
3 | 1, 2 | weq 1389 | . 2 wff x = y |
4 | wph | . . . 4 wff φ | |
5 | 4, 2 | wal 1240 | . . 3 wff ∀yφ |
6 | 3, 4 | wi 4 | . . . 4 wff (x = y → φ) |
7 | 6, 1 | wal 1240 | . . 3 wff ∀x(x = y → φ) |
8 | 5, 7 | wi 4 | . 2 wff (∀yφ → ∀x(x = y → φ)) |
9 | 3, 8 | wi 4 | 1 wff (x = y → (∀yφ → ∀x(x = y → φ))) |
Colors of variables: wff set class |
This axiom is referenced by: ax10o 1600 equs5a 1672 sbcof2 1688 ax11o 1700 ax11v 1705 |
Copyright terms: Public domain | W3C validator |