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

Theorem alexn 1641
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 1628 . . 3  |-  ( E. y  -.  ph  <->  -.  A. y ph )
21albii 1620 . 2  |-  ( A. x E. y  -.  ph  <->  A. x  -.  A. y ph )
3 alnex 1598 . 2  |-  ( A. x  -.  A. y ph  <->  -. 
E. x A. y ph )
42, 3bitri 249 1  |-  ( A. x E. y  -.  ph  <->  -. 
E. x A. y ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184   A.wal 1377   E.wex 1596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612
This theorem depends on definitions:  df-bi 185  df-ex 1597
This theorem is referenced by:  2exnexn  1642  nalset  4584  kmlem2  8530  bj-nalset  33470
  Copyright terms: Public domain W3C validator