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Theorem pm14.24 36753
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 2279 . . . . 5  |-  F/ x E! x ph
2 nfsbc1v 3319 . . . . 5  |-  F/ x [. y  /  x ]. ph
3 pm14.12 36742 . . . . . . . . . 10  |-  ( E! x ph  ->  A. x A. y ( ( ph  /\ 
[. y  /  x ]. ph )  ->  x  =  y ) )
4319.21bbi 1925 . . . . . . . . 9  |-  ( E! x ph  ->  (
( ph  /\  [. y  /  x ]. ph )  ->  x  =  y ) )
54ancomsd 455 . . . . . . . 8  |-  ( E! x ph  ->  (
( [. y  /  x ]. ph  /\  ph )  ->  x  =  y ) )
65expdimp 438 . . . . . . 7  |-  ( ( E! x ph  /\  [. y  /  x ]. ph )  ->  ( ph  ->  x  =  y ) )
7 pm13.13b 36729 . . . . . . . . 9  |-  ( (
[. y  /  x ]. ph  /\  x  =  y )  ->  ph )
87ex 435 . . . . . . . 8  |-  ( [. y  /  x ]. ph  ->  ( x  =  y  ->  ph ) )
98adantl 467 . . . . . . 7  |-  ( ( E! x ph  /\  [. y  /  x ]. ph )  ->  ( x  =  y  ->  ph )
)
106, 9impbid 193 . . . . . 6  |-  ( ( E! x ph  /\  [. y  /  x ]. ph )  ->  ( ph  <->  x  =  y ) )
1110ex 435 . . . . 5  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  ->  ( ph  <->  x  =  y ) ) )
121, 2, 11alrimd 1936 . . . 4  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  ->  A. x
( ph  <->  x  =  y
) ) )
13 iotaval 5576 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
1413eqcomd 2430 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
1512, 14syl6 34 . . 3  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  ->  y  =  ( iota x ph )
) )
16 iota4 5583 . . . 4  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
17 dfsbcq 3301 . . . 4  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  x ]. ph  <->  [. ( iota
x ph )  /  x ]. ph ) )
1816, 17syl5ibrcom 225 . . 3  |-  ( E! x ph  ->  (
y  =  ( iota
x ph )  ->  [. y  /  x ]. ph )
)
1915, 18impbid 193 . 2  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  <->  y  =  ( iota x ph )
) )
2019alrimiv 1767 1  |-  ( E! x ph  ->  A. y
( [. y  /  x ]. ph  <->  y  =  ( iota x ph )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   E!weu 2269   [.wsbc 3299   iotacio 5563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rex 2777  df-v 3082  df-sbc 3300  df-un 3441  df-sn 3999  df-pr 4001  df-uni 4220  df-iota 5565
This theorem is referenced by: (None)
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