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Theorem zfrepclf 4509
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
zfrepclf.1  |-  F/_ x A
zfrepclf.2  |-  A  e. 
_V
zfrepclf.3  |-  ( x  e.  A  ->  E. z A. y ( ph  ->  y  =  z ) )
Assertion
Ref Expression
zfrepclf  |-  E. z A. y ( y  e.  z  <->  E. x ( x  e.  A  /\  ph ) )
Distinct variable groups:    y, z, A    ph, z    x, y, z
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem zfrepclf
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 zfrepclf.2 . 2  |-  A  e. 
_V
2 zfrepclf.1 . . . . . 6  |-  F/_ x A
32nfeq2 2629 . . . . 5  |-  F/ x  v  =  A
4 eleq2 2524 . . . . . 6  |-  ( v  =  A  ->  (
x  e.  v  <->  x  e.  A ) )
5 zfrepclf.3 . . . . . 6  |-  ( x  e.  A  ->  E. z A. y ( ph  ->  y  =  z ) )
64, 5syl6bi 228 . . . . 5  |-  ( v  =  A  ->  (
x  e.  v  ->  E. z A. y (
ph  ->  y  =  z ) ) )
73, 6alrimi 1813 . . . 4  |-  ( v  =  A  ->  A. x
( x  e.  v  ->  E. z A. y
( ph  ->  y  =  z ) ) )
8 nfv 1674 . . . . 5  |-  F/ z
ph
98axrep5 4508 . . . 4  |-  ( A. x ( x  e.  v  ->  E. z A. y ( ph  ->  y  =  z ) )  ->  E. z A. y
( y  e.  z  <->  E. x ( x  e.  v  /\  ph )
) )
107, 9syl 16 . . 3  |-  ( v  =  A  ->  E. z A. y ( y  e.  z  <->  E. x ( x  e.  v  /\  ph ) ) )
114anbi1d 704 . . . . . . 7  |-  ( v  =  A  ->  (
( x  e.  v  /\  ph )  <->  ( x  e.  A  /\  ph )
) )
123, 11exbid 1822 . . . . . 6  |-  ( v  =  A  ->  ( E. x ( x  e.  v  /\  ph )  <->  E. x ( x  e.  A  /\  ph )
) )
1312bibi2d 318 . . . . 5  |-  ( v  =  A  ->  (
( y  e.  z  <->  E. x ( x  e.  v  /\  ph )
)  <->  ( y  e.  z  <->  E. x ( x  e.  A  /\  ph ) ) ) )
1413albidv 1680 . . . 4  |-  ( v  =  A  ->  ( A. y ( y  e.  z  <->  E. x ( x  e.  v  /\  ph ) )  <->  A. y
( y  e.  z  <->  E. x ( x  e.  A  /\  ph )
) ) )
1514exbidv 1681 . . 3  |-  ( v  =  A  ->  ( E. z A. y ( y  e.  z  <->  E. x
( x  e.  v  /\  ph ) )  <->  E. z A. y ( y  e.  z  <->  E. x
( x  e.  A  /\  ph ) ) ) )
1610, 15mpbid 210 . 2  |-  ( v  =  A  ->  E. z A. y ( y  e.  z  <->  E. x ( x  e.  A  /\  ph ) ) )
171, 16vtocle 3144 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 1368    = wceq 1370   E.wex 1587    e. wcel 1758   F/_wnfc 2599   _Vcvv 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3072
This theorem is referenced by:  zfrep3cl  4510  zfrep4  4511
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