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Theorem iotasbc 31280
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 3338 . 2  |-  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( y  =  ( iota x ph )  /\  ps ) )
2 iotaexeu 31279 . . . . . . 7  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
3 eueq 3257 . . . . . . 7  |-  ( ( iota x ph )  e.  _V  <->  E! y  y  =  ( iota x ph ) )
42, 3sylib 196 . . . . . 6  |-  ( E! x ph  ->  E! y  y  =  ( iota x ph ) )
5 df-eu 2272 . . . . . . 7  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
6 iotaval 5552 . . . . . . . . . 10  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
76eqcomd 2451 . . . . . . . . 9  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
87ancri 552 . . . . . . . 8  |-  ( A. x ( ph  <->  x  =  y )  ->  (
y  =  ( iota
x ph )  /\  A. x ( ph  <->  x  =  y ) ) )
98eximi 1643 . . . . . . 7  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y ( y  =  ( iota x ph )  /\  A. x
( ph  <->  x  =  y
) ) )
105, 9sylbi 195 . . . . . 6  |-  ( E! x ph  ->  E. y
( y  =  ( iota x ph )  /\  A. x ( ph  <->  x  =  y ) ) )
11 eupick 2344 . . . . . 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 661 . . . . 5  |-  ( E! x ph  ->  (
y  =  ( iota
x ph )  ->  A. x
( ph  <->  x  =  y
) ) )
1312, 7impbid1 203 . . . 4  |-  ( E! x ph  ->  (
y  =  ( iota
x ph )  <->  A. x
( ph  <->  x  =  y
) ) )
1413anbi1d 704 . . 3  |-  ( E! x ph  ->  (
( y  =  ( iota x ph )  /\  ps )  <->  ( A. x ( ph  <->  x  =  y )  /\  ps ) ) )
1514exbidv 1701 . 2  |-  ( E! x ph  ->  ( E. y ( y  =  ( iota x ph )  /\  ps )  <->  E. y
( A. x (
ph 
<->  x  =  y )  /\  ps ) ) )
161, 15syl5bb 257 1  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( A. x
( ph  <->  x  =  y
)  /\  ps )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1381    = wceq 1383   E.wex 1599    e. wcel 1804   E!weu 2268   _Vcvv 3095   [.wsbc 3313   iotacio 5539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rex 2799  df-v 3097  df-sbc 3314  df-un 3466  df-sn 4015  df-pr 4017  df-uni 4235  df-iota 5541
This theorem is referenced by:  iotasbc2  31281  iotavalb  31291  fvsb  31315
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