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Axiom ax-sep 4157
Description: The Axiom of Separation of ZF set theory. See axsep 4156 for more information. It was derived as axsep 4156 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 11-Sep-2006.)
Assertion
Ref Expression
ax-sep  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
Distinct variable groups:    x, y,
z    ph, y, z
Allowed substitution hint:    ph( x)

Detailed syntax breakdown of Axiom ax-sep
StepHypRef Expression
1 vx . . . . 5  set  x
2 vy . . . . 5  set  y
31, 2wel 1697 . . . 4  wff  x  e.  y
4 vz . . . . . 6  set  z
51, 4wel 1697 . . . . 5  wff  x  e.  z
6 wph . . . . 5  wff  ph
75, 6wa 358 . . . 4  wff  ( x  e.  z  /\  ph )
83, 7wb 176 . . 3  wff  ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
98, 1wal 1530 . 2  wff  A. x
( x  e.  y  <-> 
( x  e.  z  /\  ph ) )
109, 2wex 1531 1  wff  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
Colors of variables: wff set class
This axiom is referenced by:  axsep2  4158  zfauscl  4159  bm1.3ii  4160  ax9vsep  4161  axnul  4164  nalset  4167
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