MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alexn Unicode version

Theorem alexn 1569
Description: A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
Assertion
Ref Expression
alexn  |-  ( A. x E. y  -.  ph  <->  -. 
E. x A. y ph )

Proof of Theorem alexn
StepHypRef Expression
1 exnal 1564 . . 3  |-  ( E. y  -.  ph  <->  -.  A. y ph )
21albii 1556 . 2  |-  ( A. x E. y  -.  ph  <->  A. x  -.  A. y ph )
3 alnex 1533 . 2  |-  ( A. x  -.  A. y ph  <->  -. 
E. x A. y ph )
42, 3bitri 240 1  |-  ( A. x E. y  -.  ph  <->  -. 
E. x A. y ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   A.wal 1530   E.wex 1531
This theorem is referenced by:  2exnexn  1570  nalset  4167  kmlem2  7793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547
This theorem depends on definitions:  df-bi 177  df-ex 1532
  Copyright terms: Public domain W3C validator