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Theorem pm11.57 36739
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 1761 . . . . 5  |-  F/ y
ph
21nfal 2030 . . . 4  |-  F/ y A. x ph
3 sp 1937 . . . . 5  |-  ( A. x ph  ->  ph )
4 stdpc4 2184 . . . . 5  |-  ( A. x ph  ->  [ y  /  x ] ph )
53, 4jca 535 . . . 4  |-  ( A. x ph  ->  ( ph  /\ 
[ y  /  x ] ph ) )
62, 5alrimi 1955 . . 3  |-  ( A. x ph  ->  A. y
( ph  /\  [ y  /  x ] ph ) )
76axc4i 1980 . 2  |-  ( A. x ph  ->  A. x A. y ( ph  /\  [ y  /  x ] ph ) )
8 simpl 459 . . . 4  |-  ( (
ph  /\  [ y  /  x ] ph )  ->  ph )
98sps 1943 . . 3  |-  ( A. y ( ph  /\  [ y  /  x ] ph )  ->  ph )
109alimi 1684 . 2  |-  ( A. x A. y ( ph  /\ 
[ y  /  x ] ph )  ->  A. x ph )
117, 10impbii 191 1  |-  ( A. x ph  <->  A. x A. y
( ph  /\  [ y  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371   A.wal 1442   [wsb 1797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668  df-sb 1798
This theorem is referenced by: (None)
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