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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 1866). 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 1807, ax11v2 1800 and ax-11o 1803. (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 1483 | . 2 wff 𝑥 = 𝑦 |
4 | wph | . . . 4 wff 𝜑 | |
5 | 4, 2 | wal 1333 | . . 3 wff ∀𝑦𝜑 |
6 | 3, 4 | wi 4 | . . . 4 wff (𝑥 = 𝑦 → 𝜑) |
7 | 6, 1 | wal 1333 | . . 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 1695 equs5a 1774 sbcof2 1790 ax11o 1802 ax11v 1807 |
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