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Theorem pm14.24 27632
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 2153 . . . . 5  |-  F/ x E! x ph
2 nfsbc1v 3010 . . . . 5  |-  F/ x [. y  /  x ]. ph
3 pm14.12 27621 . . . . . . . . . 10  |-  ( E! x ph  ->  A. x A. y ( ( ph  /\ 
[. y  /  x ]. ph )  ->  x  =  y ) )
4319.21bbi 1795 . . . . . . . . 9  |-  ( E! x ph  ->  (
( ph  /\  [. y  /  x ]. ph )  ->  x  =  y ) )
54ancomsd 440 . . . . . . . 8  |-  ( E! x ph  ->  (
( [. y  /  x ]. ph  /\  ph )  ->  x  =  y ) )
65expdimp 426 . . . . . . 7  |-  ( ( E! x ph  /\  [. y  /  x ]. ph )  ->  ( ph  ->  x  =  y ) )
7 pm13.13b 27608 . . . . . . . . . 10  |-  ( (
[. y  /  x ]. ph  /\  x  =  y )  ->  ph )
87expcom 424 . . . . . . . . 9  |-  ( x  =  y  ->  ( [. y  /  x ]. ph  ->  ph ) )
98com12 27 . . . . . . . 8  |-  ( [. y  /  x ]. ph  ->  ( x  =  y  ->  ph ) )
109adantl 452 . . . . . . 7  |-  ( ( E! x ph  /\  [. y  /  x ]. ph )  ->  ( x  =  y  ->  ph )
)
116, 10impbid 183 . . . . . 6  |-  ( ( E! x ph  /\  [. y  /  x ]. ph )  ->  ( ph  <->  x  =  y ) )
1211ex 423 . . . . 5  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  ->  ( ph  <->  x  =  y ) ) )
131, 2, 12alrimd 1749 . . . 4  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  ->  A. x
( ph  <->  x  =  y
) ) )
14 iotaval 5230 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
1514eqcomd 2288 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
1613, 15syl6 29 . . 3  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  ->  y  =  ( iota x ph )
) )
17 iota4 5237 . . . 4  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
18 dfsbcq 2993 . . . 4  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  x ]. ph  <->  [. ( iota
x ph )  /  x ]. ph ) )
1917, 18syl5ibrcom 213 . . 3  |-  ( E! x ph  ->  (
y  =  ( iota
x ph )  ->  [. y  /  x ]. ph )
)
2016, 19impbid 183 . 2  |-  ( E! x ph  ->  ( [. y  /  x ]. ph  <->  y  =  ( iota x ph )
) )
2120alrimiv 1617 1  |-  ( E! x ph  ->  A. y
( [. y  /  x ]. ph  <->  y  =  ( iota x ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623   E!weu 2143   [.wsbc 2991   iotacio 5217
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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