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Theorem iotasbc 27019
Description: Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define  iota in terms of a function of  ( iota x ph ). Their definition differs in that a function of  ( iota x ph ) evaluates to "false" when there isn't a single  x that satisfies  ph. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotasbc  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( A. x
( ph  <->  x  =  y
)  /\  ps )
) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)

Proof of Theorem iotasbc
StepHypRef Expression
1 sbc5 3017 . 2  |-  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( y  =  ( iota x ph )  /\  ps ) )
2 iotaexeu 27018 . . . . . . 7  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
3 eueq 2939 . . . . . . 7  |-  ( ( iota x ph )  e.  _V  <->  E! y  y  =  ( iota x ph ) )
42, 3sylib 190 . . . . . 6  |-  ( E! x ph  ->  E! y  y  =  ( iota x ph ) )
5 df-eu 2149 . . . . . . 7  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
6 iotaval 6264 . . . . . . . . . 10  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
76eqcomd 2290 . . . . . . . . 9  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
87ancri 537 . . . . . . . 8  |-  ( A. x ( ph  <->  x  =  y )  ->  (
y  =  ( iota
x ph )  /\  A. x ( ph  <->  x  =  y ) ) )
98eximi 1564 . . . . . . 7  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y ( y  =  ( iota x ph )  /\  A. x
( ph  <->  x  =  y
) ) )
105, 9sylbi 189 . . . . . 6  |-  ( E! x ph  ->  E. y
( y  =  ( iota x ph )  /\  A. x ( ph  <->  x  =  y ) ) )
11 eupick 2208 . . . . . 6  |-  ( ( E! y  y  =  ( iota x ph )  /\  E. y ( y  =  ( iota
x ph )  /\  A. x ( ph  <->  x  =  y ) ) )  ->  ( y  =  ( iota x ph )  ->  A. x ( ph  <->  x  =  y ) ) )
124, 10, 11syl2anc 644 . . . . 5  |-  ( E! x ph  ->  (
y  =  ( iota
x ph )  ->  A. x
( ph  <->  x  =  y
) ) )
1312, 7impbid1 196 . . . 4  |-  ( E! x ph  ->  (
y  =  ( iota
x ph )  <->  A. x
( ph  <->  x  =  y
) ) )
1413anbi1d 687 . . 3  |-  ( E! x ph  ->  (
( y  =  ( iota x ph )  /\  ps )  <->  ( A. x ( ph  <->  x  =  y )  /\  ps ) ) )
1514exbidv 1613 . 2  |-  ( E! x ph  ->  ( E. y ( y  =  ( iota x ph )  /\  ps )  <->  E. y
( A. x (
ph 
<->  x  =  y )  /\  ps ) ) )
161, 15syl5bb 250 1  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( A. x
( ph  <->  x  =  y
)  /\  ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1528   E.wex 1529    = wceq 1624    e. wcel 1685   E!weu 2145   _Vcvv 2790   [.wsbc 2993   iotacio 6251
This theorem is referenced by:  iotasbc2  27020  iotavalb  27030  fvsb  27055
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-rex 2551  df-v 2792  df-sbc 2994  df-un 3159  df-sn 3648  df-pr 3649  df-uni 3830  df-iota 6253
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