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Theorem r19.9rzv 3723
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 3722 . . . 4  |-  ( A  =/=  (/)  ->  ( -.  ph  <->  A. x  e.  A  -.  ph ) )
21bicomd 194 . . 3  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  -.  ph  <->  -. 
ph ) )
32con2bid 321 . 2  |-  ( A  =/=  (/)  ->  ( ph  <->  -. 
A. x  e.  A  -.  ph ) )
4 dfrex2 2719 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
53, 4syl6bbr 256 1  |-  ( A  =/=  (/)  ->  ( ph  <->  E. x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    =/= wne 2600   A.wral 2706   E.wrex 2707   (/)c0 3629
This theorem is referenced by:  r19.45zv  3726  r19.36zv  3729  iunconst  4102  fconstfv  5955  dvdsr02  15762  voliune  24586  dya2iocuni  24634  filnetlem4  26411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-v 2959  df-dif 3324  df-nul 3630
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