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Theorem r19.9rzv 3548
Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.9rzv  |-  ( A  =/=  (/)  ->  ( ph  <->  E. x  e.  A  ph ) )
Distinct variable groups:    x, A    ph, x

Proof of Theorem r19.9rzv
StepHypRef Expression
1 r19.3rzv 3547 . . . 4  |-  ( A  =/=  (/)  ->  ( -.  ph  <->  A. x  e.  A  -.  ph ) )
21bicomd 192 . . 3  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  -.  ph  <->  -. 
ph ) )
32con2bid 319 . 2  |-  ( A  =/=  (/)  ->  ( ph  <->  -. 
A. x  e.  A  -.  ph ) )
4 dfrex2 2556 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
53, 4syl6bbr 254 1  |-  ( A  =/=  (/)  ->  ( ph  <->  E. x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    =/= wne 2446   A.wral 2543   E.wrex 2544   (/)c0 3455
This theorem is referenced by:  r19.45zv  3551  r19.36zv  3554  iunconst  3913  fconstfv  5734  dvdsr02  15438  indf1ofs  23609  filnetlem4  26330
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-rex 2549  df-v 2790  df-dif 3155  df-nul 3456
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