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Axiom ax-setind 4247
Description: Axiom of  e.-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. a ( A. y  e.  a  [ y  /  a ] ph  ->  ph )  ->  A. a ph )
Distinct variable groups:    y, a    ph, y
Allowed substitution hint:    ph( a)

Detailed syntax breakdown of Axiom ax-setind
StepHypRef Expression
1 wph . . . . . 6  wff  ph
2 va . . . . . 6  setvar  a
3 vy . . . . . 6  setvar  y
41, 2, 3wsb 1645 . . . . 5  wff  [ y  /  a ] ph
52cv 1242 . . . . 5  class  a
64, 3, 5wral 2303 . . . 4  wff  A. y  e.  a  [ y  /  a ] ph
76, 1wi 4 . . 3  wff  ( A. y  e.  a  [
y  /  a ]
ph  ->  ph )
87, 2wal 1241 . 2  wff  A. a
( A. y  e.  a  [ y  / 
a ] ph  ->  ph )
91, 2wal 1241 . 2  wff  A. a ph
108, 9wi 4 1  wff  ( A. a ( A. y  e.  a  [ y  /  a ] ph  ->  ph )  ->  A. a ph )
Colors of variables: wff set class
This axiom is referenced by:  setindel  4248  elirr  4251  en2lp  4263  tfi  4283  setindft  9954  setindis  9956
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