Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm11.58 Unicode version

Theorem pm11.58 26921
Description: Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm11.58  |-  ( E. x ph  <->  E. x E. y ( ph  /\  [ y  /  x ] ph ) )
Distinct variable group:    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem pm11.58
StepHypRef Expression
1 19.8a 1758 . . . . 5  |-  ( ph  ->  E. x ph )
2 nfv 1629 . . . . . 6  |-  F/ y
ph
32sb8e 1988 . . . . 5  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
41, 3sylib 190 . . . 4  |-  ( ph  ->  E. y [ y  /  x ] ph )
54pm4.71i 616 . . 3  |-  ( ph  <->  (
ph  /\  E. y [ y  /  x ] ph ) )
6 19.42v 2039 . . 3  |-  ( E. y ( ph  /\  [ y  /  x ] ph )  <->  ( ph  /\  E. y [ y  /  x ] ph ) )
75, 6bitr4i 245 . 2  |-  ( ph  <->  E. y ( ph  /\  [ y  /  x ] ph ) )
87exbii 1580 1  |-  ( E. x ph  <->  E. x E. y ( ph  /\  [ y  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1537   [wsb 1883
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884
  Copyright terms: Public domain W3C validator