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Theorem ralinexa 2850
Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
Assertion
Ref Expression
ralinexa  |-  ( A. x  e.  A  ( ph  ->  -.  ps )  <->  -. 
E. x  e.  A  ( ph  /\  ps )
)

Proof of Theorem ralinexa
StepHypRef Expression
1 imnan 428 . . 3  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
21ralbii 2830 . 2  |-  ( A. x  e.  A  ( ph  ->  -.  ps )  <->  A. x  e.  A  -.  ( ph  /\  ps )
)
3 ralnex 2845 . 2  |-  ( A. x  e.  A  -.  ( ph  /\  ps )  <->  -. 
E. x  e.  A  ( ph  /\  ps )
)
42, 3bitri 257 1  |-  ( A. x  e.  A  ( ph  ->  -.  ps )  <->  -. 
E. x  e.  A  ( ph  /\  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375   A.wral 2748   E.wrex 2749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-ral 2753  df-rex 2754
This theorem is referenced by:  kmlem7  8611  kmlem13  8617  lspsncv0  18417  ntreq0  20141  lhop1lem  23013  soseq  30540  ltrnnid  33745
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