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Theorem rexnalim 2459
Description: Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
rexnalim  |-  ( E. x  e.  A  -.  ph 
->  -.  A. x  e.  A  ph )

Proof of Theorem rexnalim
StepHypRef Expression
1 df-rex 2454 . 2  |-  ( E. x  e.  A  -.  ph  <->  E. x ( x  e.  A  /\  -.  ph ) )
2 exanaliim 1640 . . 3  |-  ( E. x ( x  e.  A  /\  -.  ph )  ->  -.  A. x
( x  e.  A  ->  ph ) )
3 df-ral 2453 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
42, 3sylnibr 672 . 2  |-  ( E. x ( x  e.  A  /\  -.  ph )  ->  -.  A. x  e.  A  ph )
51, 4sylbi 120 1  |-  ( E. x  e.  A  -.  ph 
->  -.  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103   A.wal 1346   E.wex 1485    e. wcel 2141   A.wral 2448   E.wrex 2449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-ral 2453  df-rex 2454
This theorem is referenced by:  nnral  2460  ralexim  2462  iundif2ss  3936  ixp0  6707  omniwomnimkv  7141  alzdvds  11807  pc2dvds  12276  isnsgrp  12640  nninfsellemeq  14012
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