MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbe1 Unicode version

Theorem hbe1 1705
Description:  x is not free in  E. x ph. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
hbe1  |-  ( E. x ph  ->  A. x E. x ph )

Proof of Theorem hbe1
StepHypRef Expression
1 df-ex 1529 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
2 hbn1 1704 . 2  |-  ( -. 
A. x  -.  ph  ->  A. x  -.  A. x  -.  ph )
31, 2hbxfrbi 1555 1  |-  ( E. x ph  ->  A. x E. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527   E.wex 1528
This theorem is referenced by:  nfe1  1706  19.23h  1728  ax12olem5  1872  ax10lem2  1877  exlimexi  28287  vk15.4j  28291  vk15.4jVD  28690  a12study5rev  29122  a12study10  29136  a12study10n  29137  ax9lem15  29154
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-6 1703
This theorem depends on definitions:  df-bi 177  df-ex 1529
  Copyright terms: Public domain W3C validator