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Theorem pm11.58 36810
Description: Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm11.58  |-  ( E. x ph  <->  E. x E. y ( ph  /\  [ y  /  x ] ph ) )
Distinct variable group:    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem pm11.58
StepHypRef Expression
1 19.8a 1955 . . . . 5  |-  ( ph  ->  E. x ph )
2 nfv 1769 . . . . . 6  |-  F/ y
ph
32sb8e 2274 . . . . 5  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
41, 3sylib 201 . . . 4  |-  ( ph  ->  E. y [ y  /  x ] ph )
54pm4.71i 644 . . 3  |-  ( ph  <->  (
ph  /\  E. y [ y  /  x ] ph ) )
6 19.42v 1842 . . 3  |-  ( E. y ( ph  /\  [ y  /  x ] ph )  <->  ( ph  /\  E. y [ y  /  x ] ph ) )
75, 6bitr4i 260 . 2  |-  ( ph  <->  E. y ( ph  /\  [ y  /  x ] ph ) )
87exbii 1726 1  |-  ( E. x ph  <->  E. x E. y ( ph  /\  [ y  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376   E.wex 1671   [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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