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Theorem r19.3rz 3545
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 3464 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 biimt 325 . . 3  |-  ( E. x  x  e.  A  ->  ( ph  <->  ( E. x  x  e.  A  ->  ph ) ) )
31, 2sylbi 187 . 2  |-  ( A  =/=  (/)  ->  ( ph  <->  ( E. x  x  e.  A  ->  ph ) ) )
4 df-ral 2548 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
5 r19.3rz.1 . . . 4  |-  F/ x ph
6519.23 1797 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  ( E. x  x  e.  A  ->  ph ) )
74, 6bitri 240 . 2  |-  ( A. x  e.  A  ph  <->  ( E. x  x  e.  A  ->  ph ) )
83, 7syl6bbr 254 1  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   E.wex 1528   F/wnf 1531    e. wcel 1684    =/= wne 2446   A.wral 2543   (/)c0 3455
This theorem is referenced by:  r19.28z  3546  r19.27z  3552  2reu4a  27967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-nul 3456
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