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Theorem pm14.24 27609
Description: Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
pm14.24  |-  ( E! x ph  ->  A. y
( [. y  /  x ]. ph  <->  y  =  ( iota x ph )
) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem pm14.24
StepHypRef Expression
1 nfeu1 2291 . . . . 5  |-  F/ x E! x ph
2 nfsbc1v 3180 . . . . 5  |-  F/ x [. y  /  x ]. ph
3 pm14.12 27598 . . . . . . . . . 10  |-  ( E! x ph  ->  A. x A. y ( ( ph  /\ 
[. y  /  x ]. ph )  ->  x  =  y ) )
4319.21bbi 1888 . . . . . . . . 9  |-  ( E! x ph  ->  (
( ph  /\  [. y  /  x ]. ph )  ->  x  =  y ) )
54ancomsd 441 . . . . . . . 8  |-  ( E! x ph  ->  (
( [. y  /  x ]. ph  /\  ph )  ->  x  =  y ) )
65expdimp 427 . . . . . . 7  |-  ( ( E! x ph  /\  [. y  /  x ]. ph )  ->  ( ph  ->  x  =  y ) )
7 pm13.13b 27585 . . . . . . . . 9  |-  ( (
[. y  /  x ]. ph  /\  x  =  y )  ->  ph )
87ex 424 . . . . . . . 8  |-  ( [. y  /  x ]. ph  ->  ( x  =  y  ->  ph ) )
98adantl 453 . . . . . . 7  |-  ( ( E! x ph  /\  [. y  /  x ]. ph )  ->  ( x  =  y  ->  ph )
)
106, 9impbid 184 . . . . . 6  |-  ( ( E! x ph  /\  [. y  /  x ]. ph )  ->  ( ph  <->  x  =  y ) )
1110ex 424 . . . . 5  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  ->  ( ph  <->  x  =  y ) ) )
121, 2, 11alrimd 1785 . . . 4  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  ->  A. x
( ph  <->  x  =  y
) ) )
13 iotaval 5429 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
1413eqcomd 2441 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
1512, 14syl6 31 . . 3  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  ->  y  =  ( iota x ph )
) )
16 iota4 5436 . . . 4  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
17 dfsbcq 3163 . . . 4  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  x ]. ph  <->  [. ( iota
x ph )  /  x ]. ph ) )
1816, 17syl5ibrcom 214 . . 3  |-  ( E! x ph  ->  (
y  =  ( iota
x ph )  ->  [. y  /  x ]. ph )
)
1915, 18impbid 184 . 2  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  <->  y  =  ( iota x ph )
) )
2019alrimiv 1641 1  |-  ( E! x ph  ->  A. y
( [. y  /  x ]. ph  <->  y  =  ( iota x ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652   E!weu 2281   [.wsbc 3161   iotacio 5416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-v 2958  df-sbc 3162  df-un 3325  df-sn 3820  df-pr 3821  df-uni 4016  df-iota 5418
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