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Axiom ax-setind 4220
 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. (Contributed by Jim Kingdon, 19-Oct-2018.)
Assertion
Ref Expression
ax-setind (𝑎(y 𝑎 [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 1642 . . . . 5 wff [y / 𝑎]φ
52cv 1241 . . . . 5 class 𝑎
64, 3, 5wral 2300 . . . 4 wff y 𝑎 [y / 𝑎]φ
76, 1wi 4 . . 3 wff (y 𝑎 [y / 𝑎]φφ)
87, 2wal 1240 . 2 wff 𝑎(y 𝑎 [y / 𝑎]φφ)
91, 2wal 1240 . 2 wff 𝑎φ
108, 9wi 4 1 wff (𝑎(y 𝑎 [y / 𝑎]φφ) → 𝑎φ)
 Colors of variables: wff set class This axiom is referenced by:  setindel  4221  elirr  4224  en2lp  4232  tfi  4248  setindft  9425  setindis  9427
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