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Theorem iotasbc5 27631
Description: Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotasbc5  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( y  =  ( iota x ph )  /\  ps ) ) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)

Proof of Theorem iotasbc5
StepHypRef Expression
1 sbc5 3015 . 2  |-  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( y  =  ( iota x ph )  /\  ps ) )
21a1i 10 1  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( y  =  ( iota x ph )  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623   E!weu 2143   [.wsbc 2991   iotacio 5217
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 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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