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Theorem dfral2 2873
Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 2874. (Revised by Wolf Lammen, 9-Dec-2019.)
Assertion
Ref Expression
dfral2  |-  ( A. x  e.  A  ph  <->  -.  E. x  e.  A  -.  ph )

Proof of Theorem dfral2
StepHypRef Expression
1 notnot 293 . . 3  |-  ( ph  <->  -. 
-.  ph )
21ralbii 2857 . 2  |-  ( A. x  e.  A  ph  <->  A. x  e.  A  -.  -.  ph )
3 ralnex 2872 . 2  |-  ( A. x  e.  A  -.  -.  ph  <->  -.  E. x  e.  A  -.  ph )
42, 3bitri 253 1  |-  ( A. x  e.  A  ph  <->  -.  E. x  e.  A  -.  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188   A.wral 2776   E.wrex 2777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1661  df-ral 2781  df-rex 2782
This theorem is referenced by:  rexnal  2874  boxcutc  7575  infssuni  7873  ac6n  8921  indstr  11233  trfil3  20899  tglowdim2ln  24692  nmobndseqi  26416  stri  27906  hstri  27914  bnj1204  29827  poimirlem1  31903
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