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Theorem r19.3rz 3662
Description: Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)
Hypothesis
Ref Expression
r19.3rz.1  |-  F/ x ph
Assertion
Ref Expression
r19.3rz  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem r19.3rz
StepHypRef Expression
1 n0 3580 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 biimt 326 . . 3  |-  ( E. x  x  e.  A  ->  ( ph  <->  ( E. x  x  e.  A  ->  ph ) ) )
31, 2sylbi 188 . 2  |-  ( A  =/=  (/)  ->  ( ph  <->  ( E. x  x  e.  A  ->  ph ) ) )
4 df-ral 2654 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
5 r19.3rz.1 . . . 4  |-  F/ x ph
6519.23 1809 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  ( E. x  x  e.  A  ->  ph ) )
74, 6bitri 241 . 2  |-  ( A. x  e.  A  ph  <->  ( E. x  x  e.  A  ->  ph ) )
83, 7syl6bbr 255 1  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546   E.wex 1547   F/wnf 1550    e. wcel 1717    =/= wne 2550   A.wral 2649   (/)c0 3571
This theorem is referenced by:  r19.28z  3663  r19.27z  3669  2reu4a  27635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-v 2901  df-dif 3266  df-nul 3572
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