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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 1859). 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 1800, ax11v2 1793 and ax-11o 1796. (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 1480 | . 2 wff 𝑥 = 𝑦 |
4 | wph | . . . 4 wff 𝜑 | |
5 | 4, 2 | wal 1330 | . . 3 wff ∀𝑦𝜑 |
6 | 3, 4 | wi 4 | . . . 4 wff (𝑥 = 𝑦 → 𝜑) |
7 | 6, 1 | wal 1330 | . . 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 1694 equs5a 1767 sbcof2 1783 ax11o 1795 ax11v 1800 |
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