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Theorem pm14.24 36853
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 2329 . . . . 5  |-  F/ x E! x ph
2 nfsbc1v 3275 . . . . 5  |-  F/ x [. y  /  x ]. ph
3 pm14.12 36842 . . . . . . . . . 10  |-  ( E! x ph  ->  A. x A. y ( ( ph  /\ 
[. y  /  x ]. ph )  ->  x  =  y ) )
4319.21bbi 1968 . . . . . . . . 9  |-  ( E! x ph  ->  (
( ph  /\  [. y  /  x ]. ph )  ->  x  =  y ) )
54ancomsd 461 . . . . . . . 8  |-  ( E! x ph  ->  (
( [. y  /  x ]. ph  /\  ph )  ->  x  =  y ) )
65expdimp 444 . . . . . . 7  |-  ( ( E! x ph  /\  [. y  /  x ]. ph )  ->  ( ph  ->  x  =  y ) )
7 pm13.13b 36829 . . . . . . . . 9  |-  ( (
[. y  /  x ]. ph  /\  x  =  y )  ->  ph )
87ex 441 . . . . . . . 8  |-  ( [. y  /  x ]. ph  ->  ( x  =  y  ->  ph ) )
98adantl 473 . . . . . . 7  |-  ( ( E! x ph  /\  [. y  /  x ]. ph )  ->  ( x  =  y  ->  ph )
)
106, 9impbid 195 . . . . . 6  |-  ( ( E! x ph  /\  [. y  /  x ]. ph )  ->  ( ph  <->  x  =  y ) )
1110ex 441 . . . . 5  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  ->  ( ph  <->  x  =  y ) ) )
121, 2, 11alrimd 1979 . . . 4  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  ->  A. x
( ph  <->  x  =  y
) ) )
13 iotaval 5564 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
1413eqcomd 2477 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
1512, 14syl6 33 . . 3  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  ->  y  =  ( iota x ph )
) )
16 iota4 5571 . . . 4  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
17 dfsbcq 3257 . . . 4  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  x ]. ph  <->  [. ( iota
x ph )  /  x ]. ph ) )
1816, 17syl5ibrcom 230 . . 3  |-  ( E! x ph  ->  (
y  =  ( iota
x ph )  ->  [. y  /  x ]. ph )
)
1915, 18impbid 195 . 2  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  <->  y  =  ( iota x ph )
) )
2019alrimiv 1781 1  |-  ( E! x ph  ->  A. y
( [. y  /  x ]. ph  <->  y  =  ( iota x ph )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450    = wceq 1452   E!weu 2319   [.wsbc 3255   iotacio 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-v 3033  df-sbc 3256  df-un 3395  df-sn 3960  df-pr 3962  df-uni 4191  df-iota 5553
This theorem is referenced by: (None)
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