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Theorem ralnex 2359
Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
Assertion
Ref Expression
ralnex  |-  ( A. x  e.  A  -.  ph  <->  -. 
E. x  e.  A  ph )

Proof of Theorem ralnex
StepHypRef Expression
1 df-ral 2354 . 2  |-  ( A. x  e.  A  -.  ph  <->  A. x ( x  e.  A  ->  -.  ph )
)
2 alinexa 1535 . . 3  |-  ( A. x ( x  e.  A  ->  -.  ph )  <->  -. 
E. x ( x  e.  A  /\  ph ) )
3 df-rex 2355 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
42, 3xchbinxr 641 . 2  |-  ( A. x ( x  e.  A  ->  -.  ph )  <->  -. 
E. x  e.  A  ph )
51, 4bitri 182 1  |-  ( A. x  e.  A  -.  ph  <->  -. 
E. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283   E.wex 1422    e. wcel 1434   A.wral 2349   E.wrex 2350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie2 1424
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-ral 2354  df-rex 2355
This theorem is referenced by:  rexalim  2362  ralinexa  2394  nrex  2454  nrexdv  2455  uni0b  3634  iindif2m  3753  supmoti  6465  suprnubex  8098  icc0r  9025  ioo0  9346  ico0  9348  ioc0  9349  prmind2  10646  sqrt2irr  10685
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