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Theorem pm11.57 26920
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 1629 . . . . 5  |-  F/ y
ph
21nfal 1732 . . . 4  |-  F/ y A. x ph
3 ax-4 1692 . . . . 5  |-  ( A. x ph  ->  ph )
4 stdpc4 1897 . . . . 5  |-  ( A. x ph  ->  [ y  /  x ] ph )
53, 4jca 520 . . . 4  |-  ( A. x ph  ->  ( ph  /\ 
[ y  /  x ] ph ) )
62, 5alrimi 1706 . . 3  |-  ( A. x ph  ->  A. y
( ph  /\  [ y  /  x ] ph ) )
76a5i 1721 . 2  |-  ( A. x ph  ->  A. x A. y ( ph  /\  [ y  /  x ] ph ) )
8 simpl 445 . . . 4  |-  ( (
ph  /\  [ y  /  x ] ph )  ->  ph )
98a4s 1700 . . 3  |-  ( A. y ( ph  /\  [ y  /  x ] ph )  ->  ph )
109alimi 1546 . 2  |-  ( A. x A. y ( ph  /\ 
[ y  /  x ] ph )  ->  A. x ph )
117, 10impbii 182 1  |-  ( A. x ph  <->  A. x A. y
( ph  /\  [ y  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   A.wal 1532   [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-17 1628  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540  df-sb 1884
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