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Theorem sbaniota 27297
Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
sbaniota  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  <->  [. ( iota x ph )  /  x ]. ps ) )

Proof of Theorem sbaniota
StepHypRef Expression
1 eupickbi 2297 . 2  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )
2 sbiota1 27296 . 2  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  <->  [. ( iota x ph )  /  x ]. ps ) )
31, 2bitrd 245 1  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  <->  [. ( iota x ph )  /  x ]. ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547   E!weu 2231   [.wsbc 3097   iotacio 5349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-v 2894  df-sbc 3098  df-un 3261  df-sn 3756  df-pr 3757  df-uni 3951  df-iota 5351
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