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Theorem iotasbc 27619
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 3015 . 2  |-  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( y  =  ( iota x ph )  /\  ps ) )
2 iotaexeu 27618 . . . . . . 7  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
3 eueq 2937 . . . . . . 7  |-  ( ( iota x ph )  e.  _V  <->  E! y  y  =  ( iota x ph ) )
42, 3sylib 188 . . . . . 6  |-  ( E! x ph  ->  E! y  y  =  ( iota x ph ) )
5 df-eu 2147 . . . . . . 7  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
6 iotaval 5230 . . . . . . . . . 10  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
76eqcomd 2288 . . . . . . . . 9  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
87ancri 535 . . . . . . . 8  |-  ( A. x ( ph  <->  x  =  y )  ->  (
y  =  ( iota
x ph )  /\  A. x ( ph  <->  x  =  y ) ) )
98eximi 1563 . . . . . . 7  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y ( y  =  ( iota x ph )  /\  A. x
( ph  <->  x  =  y
) ) )
105, 9sylbi 187 . . . . . 6  |-  ( E! x ph  ->  E. y
( y  =  ( iota x ph )  /\  A. x ( ph  <->  x  =  y ) ) )
11 eupick 2206 . . . . . 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 642 . . . . 5  |-  ( E! x ph  ->  (
y  =  ( iota
x ph )  ->  A. x
( ph  <->  x  =  y
) ) )
1312, 7impbid1 194 . . . 4  |-  ( E! x ph  ->  (
y  =  ( iota
x ph )  <->  A. x
( ph  <->  x  =  y
) ) )
1413anbi1d 685 . . 3  |-  ( E! x ph  ->  (
( y  =  ( iota x ph )  /\  ps )  <->  ( A. x ( ph  <->  x  =  y )  /\  ps ) ) )
1514exbidv 1612 . 2  |-  ( E! x ph  ->  ( E. y ( y  =  ( iota x ph )  /\  ps )  <->  E. y
( A. x (
ph 
<->  x  =  y )  /\  ps ) ) )
161, 15syl5bb 248 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 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   _Vcvv 2788   [.wsbc 2991   iotacio 5217
This theorem is referenced by:  iotasbc2  27620  iotavalb  27630  fvsb  27655
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-sbc 2992  df-un 3157  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219
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