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Axiom ax-setind 4315
Description: Axiom of -Induction (also known as set induction). An axiom of Intuitionistic Zermelo-Fraenkel set theory. Axiom 9 of [Crosilla] p. "Axioms of CZF and IZF". This replaces the Axiom of Foundation (also called Regularity) from Zermelo-Fraenkel set theory.

For more on axioms which might be adopted which are incompatible with this axiom (that is, Non-wellfounded Set Theory but in the absence of excluded middle), see Chapter 20 of [AczelRathjen], p. 183. (Contributed by Jim Kingdon, 19-Oct-2018.)

Assertion
Ref Expression
ax-setind (∀𝑎(∀𝑦𝑎 [𝑦 / 𝑎]𝜑𝜑) → ∀𝑎𝜑)
Distinct variable groups:   𝑦,𝑎   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑎)

Detailed syntax breakdown of Axiom ax-setind
StepHypRef Expression
1 wph . . . . . 6 wff 𝜑
2 va . . . . . 6 setvar 𝑎
3 vy . . . . . 6 setvar 𝑦
41, 2, 3wsb 1687 . . . . 5 wff [𝑦 / 𝑎]𝜑
52cv 1284 . . . . 5 class 𝑎
64, 3, 5wral 2353 . . . 4 wff 𝑦𝑎 [𝑦 / 𝑎]𝜑
76, 1wi 4 . . 3 wff (∀𝑦𝑎 [𝑦 / 𝑎]𝜑𝜑)
87, 2wal 1283 . 2 wff 𝑎(∀𝑦𝑎 [𝑦 / 𝑎]𝜑𝜑)
91, 2wal 1283 . 2 wff 𝑎𝜑
108, 9wi 4 1 wff (∀𝑎(∀𝑦𝑎 [𝑦 / 𝑎]𝜑𝜑) → ∀𝑎𝜑)
Colors of variables: wff set class
This axiom is referenced by:  setindel  4316  elirr  4319  en2lp  4332  zfregfr  4351  tfi  4359  setindft  11202  setindis  11204
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