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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 1886). 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 1827, ax11v2 1820 and ax-11o 1823. (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 1503 | . 2 wff 𝑥 = 𝑦 |
4 | wph | . . . 4 wff 𝜑 | |
5 | 4, 2 | wal 1351 | . . 3 wff ∀𝑦𝜑 |
6 | 3, 4 | wi 4 | . . . 4 wff (𝑥 = 𝑦 → 𝜑) |
7 | 6, 1 | wal 1351 | . . 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 1715 equs5a 1794 sbcof2 1810 ax11o 1822 ax11v 1827 |
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