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Theorem dfral2 2839
Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) Allow shortening of rexnal 2840. (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 289 . . 3  |-  ( ph  <->  -. 
-.  ph )
21ralbii 2823 . 2  |-  ( A. x  e.  A  ph  <->  A. x  e.  A  -.  -.  ph )
3 ralnex 2838 . 2  |-  ( A. x  e.  A  -.  -.  ph  <->  -.  E. x  e.  A  -.  ph )
42, 3bitri 249 1  |-  ( A. x  e.  A  ph  <->  -.  E. x  e.  A  -.  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184   A.wral 2742   E.wrex 2743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1628  df-ral 2747  df-rex 2748
This theorem is referenced by:  rexnal  2840  boxcutc  7449  infssuni  7744  ac6n  8796  indstr  11087  trfil3  20493  tglowdim2ln  24173  nmobndseqi  25832  stri  27313  hstri  27321  bnj1204  34450
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