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Theorem zfrep3cl 4565
Description: An inference rule based on the Axiom of Replacement. Typically,  ph defines a function from  x to  y. (Contributed by NM, 26-Nov-1995.)
Hypotheses
Ref Expression
zfrep3cl.1  |-  A  e. 
_V
zfrep3cl.2  |-  ( x  e.  A  ->  E. z A. y ( ph  ->  y  =  z ) )
Assertion
Ref Expression
zfrep3cl  |-  E. z A. y ( y  e.  z  <->  E. x ( x  e.  A  /\  ph ) )
Distinct variable groups:    x, y,
z, A    ph, z
Allowed substitution hints:    ph( x, y)

Proof of Theorem zfrep3cl
StepHypRef Expression
1 nfcv 2629 . 2  |-  F/_ x A
2 zfrep3cl.1 . 2  |-  A  e. 
_V
3 zfrep3cl.2 . 2  |-  ( x  e.  A  ->  E. z A. y ( ph  ->  y  =  z ) )
41, 2, 3zfrepclf 4564 1  |-  E. z A. y ( y  e.  z  <->  E. x ( x  e.  A  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377   E.wex 1596    e. wcel 1767   _Vcvv 3113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115
This theorem is referenced by: (None)
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