Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm11.57 Structured version   Unicode version

Theorem pm11.57 31539
Description: Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm11.57  |-  ( A. x ph  <->  A. x A. y
( ph  /\  [ y  /  x ] ph ) )
Distinct variable group:    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem pm11.57
StepHypRef Expression
1 nfv 1712 . . . . 5  |-  F/ y
ph
21nfal 1952 . . . 4  |-  F/ y A. x ph
3 sp 1864 . . . . 5  |-  ( A. x ph  ->  ph )
4 stdpc4 2096 . . . . 5  |-  ( A. x ph  ->  [ y  /  x ] ph )
53, 4jca 530 . . . 4  |-  ( A. x ph  ->  ( ph  /\ 
[ y  /  x ] ph ) )
62, 5alrimi 1882 . . 3  |-  ( A. x ph  ->  A. y
( ph  /\  [ y  /  x ] ph ) )
76axc4i 1903 . 2  |-  ( A. x ph  ->  A. x A. y ( ph  /\  [ y  /  x ] ph ) )
8 simpl 455 . . . 4  |-  ( (
ph  /\  [ y  /  x ] ph )  ->  ph )
98sps 1870 . . 3  |-  ( A. y ( ph  /\  [ y  /  x ] ph )  ->  ph )
109alimi 1638 . 2  |-  ( A. x A. y ( ph  /\ 
[ y  /  x ] ph )  ->  A. x ph )
117, 10impbii 188 1  |-  ( A. x ph  <->  A. x A. y
( ph  /\  [ y  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367   A.wal 1396   [wsb 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator