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Theorem r19.9rzv 3561
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 3560 . . . 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 2569 . 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 2459   A.wral 2556   E.wrex 2557   (/)c0 3468
This theorem is referenced by:  r19.45zv  3564  r19.36zv  3567  iunconst  3929  fconstfv  5750  dvdsr02  15454  indf1ofs  23624  filnetlem4  26433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-nul 3469
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