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Theorem pm11.58 31537
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 1862 . . . . 5  |-  ( ph  ->  E. x ph )
2 nfv 1712 . . . . . 6  |-  F/ y
ph
32sb8e 2170 . . . . 5  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
41, 3sylib 196 . . . 4  |-  ( ph  ->  E. y [ y  /  x ] ph )
54pm4.71i 630 . . 3  |-  ( ph  <->  (
ph  /\  E. y [ y  /  x ] ph ) )
6 19.42v 1780 . . 3  |-  ( E. y ( ph  /\  [ y  /  x ] ph )  <->  ( ph  /\  E. y [ y  /  x ] ph ) )
75, 6bitr4i 252 . 2  |-  ( ph  <->  E. y ( ph  /\  [ y  /  x ] ph ) )
87exbii 1672 1  |-  ( E. x ph  <->  E. x E. y ( ph  /\  [ y  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367   E.wex 1617   [wsb 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745
This theorem is referenced by: (None)
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