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Theorem sbaniota 27603
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 2346 . 2  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )
2 sbiota1 27602 . 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 1549   E.wex 1550   E!weu 2280   [.wsbc 3153   iotacio 5408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-sbc 3154  df-un 3317  df-sn 3812  df-pr 3813  df-uni 4008  df-iota 5410
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