MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm11.53 Structured version   Unicode version

Theorem pm11.53 1916
Description: Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm11.53  |-  ( A. x A. y ( ph  ->  ps )  <->  ( E. x ph  ->  A. y ps ) )
Distinct variable groups:    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem pm11.53
StepHypRef Expression
1 19.21v 1913 . . 3  |-  ( A. y ( ph  ->  ps )  <->  ( ph  ->  A. y ps ) )
21albii 1575 . 2  |-  ( A. x A. y ( ph  ->  ps )  <->  A. x
( ph  ->  A. y ps ) )
3 nfv 1629 . . . 4  |-  F/ x ps
43nfal 1864 . . 3  |-  F/ x A. y ps
5419.23 1819 . 2  |-  ( A. x ( ph  ->  A. y ps )  <->  ( E. x ph  ->  A. y ps ) )
62, 5bitri 241 1  |-  ( A. x A. y ( ph  ->  ps )  <->  ( E. x ph  ->  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551  df-nf 1554
  Copyright terms: Public domain W3C validator