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Theorem sbaniota 27507
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 2324 . 2  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )
2 sbiota1 27506 . 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 2258   [.wsbc 3125   iotacio 5379
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-v 2922  df-sbc 3126  df-un 3289  df-sn 3784  df-pr 3785  df-uni 3980  df-iota 5381
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