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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 1766). 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 1708, ax11v2 1701 and ax-11o 1704. (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 1392 | . 2 wff 𝑥 = 𝑦 |
4 | wph | . . . 4 wff 𝜑 | |
5 | 4, 2 | wal 1241 | . . 3 wff ∀𝑦𝜑 |
6 | 3, 4 | wi 4 | . . . 4 wff (𝑥 = 𝑦 → 𝜑) |
7 | 6, 1 | wal 1241 | . . 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 1603 equs5a 1675 sbcof2 1691 ax11o 1703 ax11v 1708 |
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