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Theorem r19.3rzv 3873
Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)
Assertion
Ref Expression
r19.3rzv  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
Distinct variable groups:    x, A    ph, x

Proof of Theorem r19.3rzv
StepHypRef Expression
1 n0 3746 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 biimt 335 . . 3  |-  ( E. x  x  e.  A  ->  ( ph  <->  ( E. x  x  e.  A  ->  ph ) ) )
31, 2sylbi 195 . 2  |-  ( A  =/=  (/)  ->  ( ph  <->  ( E. x  x  e.  A  ->  ph ) ) )
4 df-ral 2800 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
5 19.23v 1919 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  ( E. x  x  e.  A  ->  ph ) )
64, 5bitri 249 . 2  |-  ( A. x  e.  A  ph  <->  ( E. x  x  e.  A  ->  ph ) )
73, 6syl6bbr 263 1  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368   E.wex 1587    e. wcel 1758    =/= wne 2644   A.wral 2795   (/)c0 3737
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  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-ne 2646  df-ral 2800  df-v 3072  df-dif 3431  df-nul 3738
This theorem is referenced by:  r19.9rzv  3874  r19.28zv  3875  r19.37zv  3876  r19.27zv  3879  iinconst  4280  cnvpo  5475  supicc  11536  coe1mul2lem1  17830  neipeltop  18851  utop3cls  19944  rencldnfi  29300  ralnralall  30258  frgrareg  30850  frgraregord013  30851
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