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Axiom ax-bdsep 11421
Description: Axiom scheme of bounded (or restricted, or Δ0) separation. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. For the full axiom of separation, see ax-sep 3949. (Contributed by BJ, 5-Oct-2019.)
Hypothesis
Ref Expression
ax-bdsep.1  |- BOUNDED  ph
Assertion
Ref Expression
ax-bdsep  |-  A. a E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
Distinct variable groups:    a, b, x    ph, a, b
Allowed substitution hint:    ph( x)

Detailed syntax breakdown of Axiom ax-bdsep
StepHypRef Expression
1 vx . . . . . 6  setvar  x
2 vb . . . . . 6  setvar  b
31, 2wel 1439 . . . . 5  wff  x  e.  b
4 va . . . . . . 7  setvar  a
51, 4wel 1439 . . . . . 6  wff  x  e.  a
6 wph . . . . . 6  wff  ph
75, 6wa 102 . . . . 5  wff  ( x  e.  a  /\  ph )
83, 7wb 103 . . . 4  wff  ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
98, 1wal 1287 . . 3  wff  A. x
( x  e.  b  <-> 
( x  e.  a  /\  ph ) )
109, 2wex 1426 . 2  wff  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
1110, 4wal 1287 1  wff  A. a E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
Colors of variables: wff set class
This axiom is referenced by:  bdsep1  11422
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