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Theorem r19.3rz 3711
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 3629 . . 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 2702 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
5 r19.3rz.1 . . . 4  |-  F/ x ph
6519.23 1819 . . 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 1549   E.wex 1550   F/wnf 1553    e. wcel 1725    =/= wne 2598   A.wral 2697   (/)c0 3620
This theorem is referenced by:  r19.28z  3712  r19.27z  3718  2reu4a  27934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-v 2950  df-dif 3315  df-nul 3621
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