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Theorem pm11.59 36811
Description: Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
pm11.59  |-  ( A. x ( ph  ->  ps )  ->  A. y A. x ( ( ph  /\ 
[ y  /  x ] ph )  ->  ( ps  /\  [ y  /  x ] ps ) ) )
Distinct variable groups:    ph, y    ps, y
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem pm11.59
StepHypRef Expression
1 nfv 1769 . . 3  |-  F/ y ( ph  ->  ps )
21nfal 2049 . 2  |-  F/ y A. x ( ph  ->  ps )
3 sp 1957 . . . 4  |-  ( A. x ( ph  ->  ps )  ->  ( ph  ->  ps ) )
4 spsbim 2243 . . . 4  |-  ( A. x ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) )
53, 4anim12d 572 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( ( ph  /\  [ y  /  x ] ph )  -> 
( ps  /\  [
y  /  x ] ps ) ) )
65axc4i 2000 . 2  |-  ( A. x ( ph  ->  ps )  ->  A. x
( ( ph  /\  [ y  /  x ] ph )  ->  ( ps 
/\  [ y  /  x ] ps ) ) )
72, 6alrimi 1975 1  |-  ( A. x ( ph  ->  ps )  ->  A. y A. x ( ( ph  /\ 
[ y  /  x ] ph )  ->  ( ps  /\  [ y  /  x ] ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376   A.wal 1450   [wsb 1805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806
This theorem is referenced by: (None)
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