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Theorem rexanali 2674
Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
Assertion
Ref Expression
rexanali  |-  ( E. x  e.  A  (
ph  /\  -.  ps )  <->  -. 
A. x  e.  A  ( ph  ->  ps )
)

Proof of Theorem rexanali
StepHypRef Expression
1 annim 414 . . 3  |-  ( (
ph  /\  -.  ps )  <->  -.  ( ph  ->  ps ) )
21rexbii 2653 . 2  |-  ( E. x  e.  A  (
ph  /\  -.  ps )  <->  E. x  e.  A  -.  ( ph  ->  ps )
)
3 rexnal 2639 . 2  |-  ( E. x  e.  A  -.  ( ph  ->  ps )  <->  -. 
A. x  e.  A  ( ph  ->  ps )
)
42, 3bitri 240 1  |-  ( E. x  e.  A  (
ph  /\  -.  ps )  <->  -. 
A. x  e.  A  ( ph  ->  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wral 2628   E.wrex 2629
This theorem is referenced by:  dfsup2OLD  7343  qsqueeze  10680  elcls  17027  ist1-2  17292  haust1  17297  t1sep  17315  1stccnp  17405  filufint  17828  fclscf  17933  pmltpc  19025  ovolgelb  19054  itg2seq  19312  radcnvlt1  20012  pntlem3  20981  usgra2edg1  21080  wfi  24948  frind  24984  limsucncmpi  25626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-11 1751
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-ral 2633  df-rex 2634
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