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Theorem pm11.57 27456
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 1626 . . . . 5  |-  F/ y
ph
21nfal 1860 . . . 4  |-  F/ y A. x ph
3 sp 1759 . . . . 5  |-  ( A. x ph  ->  ph )
4 stdpc4 2073 . . . . 5  |-  ( A. x ph  ->  [ y  /  x ] ph )
53, 4jca 519 . . . 4  |-  ( A. x ph  ->  ( ph  /\ 
[ y  /  x ] ph ) )
62, 5alrimi 1777 . . 3  |-  ( A. x ph  ->  A. y
( ph  /\  [ y  /  x ] ph ) )
76a5i 1803 . 2  |-  ( A. x ph  ->  A. x A. y ( ph  /\  [ y  /  x ] ph ) )
8 simpl 444 . . . 4  |-  ( (
ph  /\  [ y  /  x ] ph )  ->  ph )
98sps 1766 . . 3  |-  ( A. y ( ph  /\  [ y  /  x ] ph )  ->  ph )
109alimi 1565 . 2  |-  ( A. x A. y ( ph  /\ 
[ y  /  x ] ph )  ->  A. x ph )
117, 10impbii 181 1  |-  ( A. x ph  <->  A. x A. y
( ph  /\  [ y  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   A.wal 1546   [wsb 1655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-sb 1656
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