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Axiom ax-bdsetind 10906
Description: Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
Hypothesis
Ref Expression
ax-bdsetind.bd  |- BOUNDED  ph
Assertion
Ref Expression
ax-bdsetind  |-  ( 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-bdsetind
StepHypRef Expression
1 wph . . . . . 6  wff  ph
2 va . . . . . 6  setvar  a
3 vy . . . . . 6  setvar  y
41, 2, 3wsb 1686 . . . . 5  wff  [ y  /  a ] ph
52cv 1284 . . . . 5  class  a
64, 3, 5wral 2349 . . . 4  wff  A. y  e.  a  [ y  /  a ] ph
76, 1wi 4 . . 3  wff  ( A. y  e.  a  [
y  /  a ]
ph  ->  ph )
87, 2wal 1283 . 2  wff  A. a
( A. y  e.  a  [ y  / 
a ] ph  ->  ph )
91, 2wal 1283 . 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:  bdsetindis  10907
  Copyright terms: Public domain W3C validator