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Theorem ralnex 2465
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 2460 . 2  |-  ( A. x  e.  A  -.  ph  <->  A. x ( x  e.  A  ->  -.  ph )
)
2 alinexa 1603 . . 3  |-  ( A. x ( x  e.  A  ->  -.  ph )  <->  -. 
E. x ( x  e.  A  /\  ph ) )
3 df-rex 2461 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
42, 3xchbinxr 683 . 2  |-  ( A. x ( x  e.  A  ->  -.  ph )  <->  -. 
E. x  e.  A  ph )
51, 4bitri 184 1  |-  ( A. x  e.  A  -.  ph  <->  -. 
E. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1351   E.wex 1492    e. wcel 2148   A.wral 2455   E.wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie2 1494
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-ral 2460  df-rex 2461
This theorem is referenced by:  nnral  2467  rexalim  2470  ralinexa  2504  nrex  2569  nrexdv  2570  ralnex2  2616  r19.30dc  2624  uni0b  3836  iindif2m  3956  f0rn0  5412  supmoti  6994  fodjuomnilemdc  7144  ismkvnex  7155  nninfwlpoimlemginf  7176  suprnubex  8912  icc0r  9928  ioo0  10262  ico0  10264  ioc0  10265  prmind2  12122  sqrt2irr  12164  nconstwlpolem  14851
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