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Theorem pm11.57 36594
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 1751 . . . . 5  |-  F/ y
ph
21nfal 2003 . . . 4  |-  F/ y A. x ph
3 sp 1910 . . . . 5  |-  ( A. x ph  ->  ph )
4 stdpc4 2147 . . . . 5  |-  ( A. x ph  ->  [ y  /  x ] ph )
53, 4jca 534 . . . 4  |-  ( A. x ph  ->  ( ph  /\ 
[ y  /  x ] ph ) )
62, 5alrimi 1928 . . 3  |-  ( A. x ph  ->  A. y
( ph  /\  [ y  /  x ] ph ) )
76axc4i 1953 . 2  |-  ( A. x ph  ->  A. x A. y ( ph  /\  [ y  /  x ] ph ) )
8 simpl 458 . . . 4  |-  ( (
ph  /\  [ y  /  x ] ph )  ->  ph )
98sps 1916 . . 3  |-  ( A. y ( ph  /\  [ y  /  x ] ph )  ->  ph )
109alimi 1680 . 2  |-  ( A. x A. y ( ph  /\ 
[ y  /  x ] ph )  ->  A. x ph )
117, 10impbii 190 1  |-  ( A. x ph  <->  A. x A. y
( ph  /\  [ y  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370   A.wal 1435   [wsb 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-sb 1787
This theorem is referenced by: (None)
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