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Theorem pm11.58 27000
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 1720 . . . . 5  |-  ( ph  ->  E. x ph )
2 nfv 1605 . . . . . 6  |-  F/ y
ph
32sb8e 2032 . . . . 5  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
41, 3sylib 188 . . . 4  |-  ( ph  ->  E. y [ y  /  x ] ph )
54pm4.71i 613 . . 3  |-  ( ph  <->  (
ph  /\  E. y [ y  /  x ] ph ) )
6 19.42v 1848 . . 3  |-  ( E. y ( ph  /\  [ y  /  x ] ph )  <->  ( ph  /\  E. y [ y  /  x ] ph ) )
75, 6bitr4i 243 . 2  |-  ( ph  <->  E. y ( ph  /\  [ y  /  x ] ph ) )
87exbii 1569 1  |-  ( E. x ph  <->  E. x E. y ( ph  /\  [ y  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528   [wsb 1630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631
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