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Theorem r19.3rz 3908
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 3793 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 biimt 333 . . 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 2809 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
5 r19.3rz.1 . . . 4  |-  F/ x ph
6519.23 1915 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  ( E. x  x  e.  A  ->  ph ) )
74, 6bitri 249 . 2  |-  ( A. x  e.  A  ph  <->  ( E. x  x  e.  A  ->  ph ) )
83, 7syl6bbr 263 1  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1396   E.wex 1617   F/wnf 1621    e. wcel 1823    =/= wne 2649   A.wral 2804   (/)c0 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-v 3108  df-dif 3464  df-nul 3784
This theorem is referenced by:  r19.28z  3909  r19.27z  3916  2reu4a  32436
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