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Theorem sbiota1 27033
Description: Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
sbiota1  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  <->  [. ( iota x ph )  /  x ]. ps ) )
Dummy variable  y is distinct from all other variables.

Proof of Theorem sbiota1
StepHypRef Expression
1 df-eu 2148 . . . 4  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
21biimpi 188 . . 3  |-  ( E! x ph  ->  E. y A. x ( ph  <->  x  =  y ) )
3 iota4 6270 . . 3  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
4 iotaval 6263 . . . . . 6  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
54eqcomd 2289 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
6 spsbim 2015 . . . . . . . 8  |-  ( A. x ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) )
7 sbsbc 2996 . . . . . . . 8  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
8 sbsbc 2996 . . . . . . . 8  |-  ( [ y  /  x ] ps 
<-> 
[. y  /  x ]. ps )
96, 7, 83imtr3g 262 . . . . . . 7  |-  ( A. x ( ph  ->  ps )  ->  ( [. y  /  x ]. ph  ->  [. y  /  x ]. ps ) )
10 dfsbcq 2994 . . . . . . . 8  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  x ]. ph  <->  [. ( iota
x ph )  /  x ]. ph ) )
11 dfsbcq 2994 . . . . . . . 8  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  x ]. ps  <->  [. ( iota x ph )  /  x ]. ps ) )
1210, 11imbi12d 313 . . . . . . 7  |-  ( y  =  ( iota x ph )  ->  ( (
[. y  /  x ]. ph  ->  [. y  /  x ]. ps )  <->  ( [. ( iota x ph )  /  x ]. ph  ->  [. ( iota x ph )  /  x ]. ps ) ) )
139, 12syl5ib 212 . . . . . 6  |-  ( y  =  ( iota x ph )  ->  ( A. x ( ph  ->  ps )  ->  ( [. ( iota x ph )  /  x ]. ph  ->  [. ( iota x ph )  /  x ]. ps ) ) )
1413com23 74 . . . . 5  |-  ( y  =  ( iota x ph )  ->  ( [. ( iota x ph )  /  x ]. ph  ->  ( A. x ( ph  ->  ps )  ->  [. ( iota x ph )  /  x ]. ps ) ) )
155, 14syl 17 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( [. ( iota x ph )  /  x ]. ph  ->  ( A. x ( ph  ->  ps )  ->  [. ( iota x ph )  /  x ]. ps ) ) )
1615exlimiv 1667 . . 3  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  ( [. ( iota x ph )  /  x ]. ph  ->  ( A. x ( ph  ->  ps )  ->  [. ( iota
x ph )  /  x ]. ps ) ) )
172, 3, 16sylc 58 . 2  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  ->  [. ( iota
x ph )  /  x ]. ps ) )
18 iotaexeu 27017 . . . . 5  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
1910, 11anbi12d 693 . . . . . . . 8  |-  ( y  =  ( iota x ph )  ->  ( (
[. y  /  x ]. ph  /\  [. y  /  x ]. ps )  <->  (
[. ( iota x ph )  /  x ]. ph  /\  [. ( iota x ph )  /  x ]. ps ) ) )
2019imbi1d 310 . . . . . . 7  |-  ( y  =  ( iota x ph )  ->  ( ( ( [. y  /  x ]. ph  /\  [. y  /  x ]. ps )  ->  E. x ( ph  /\ 
ps ) )  <->  ( ( [. ( iota x ph )  /  x ]. ph  /\  [. ( iota x ph )  /  x ]. ps )  ->  E. x ( ph  /\ 
ps ) ) ) )
21 sbcan 3034 . . . . . . . 8  |-  ( [. y  /  x ]. ( ph  /\  ps )  <->  ( [. y  /  x ]. ph  /\  [. y  /  x ]. ps ) )
22 spesbc 3073 . . . . . . . 8  |-  ( [. y  /  x ]. ( ph  /\  ps )  ->  E. x ( ph  /\  ps ) )
2321, 22sylbir 206 . . . . . . 7  |-  ( (
[. y  /  x ]. ph  /\  [. y  /  x ]. ps )  ->  E. x ( ph  /\ 
ps ) )
2420, 23vtoclg 2844 . . . . . 6  |-  ( ( iota x ph )  e.  _V  ->  ( ( [. ( iota x ph )  /  x ]. ph  /\  [. ( iota x ph )  /  x ]. ps )  ->  E. x ( ph  /\ 
ps ) ) )
2524exp3a 427 . . . . 5  |-  ( ( iota x ph )  e.  _V  ->  ( [. ( iota x ph )  /  x ]. ph  ->  (
[. ( iota x ph )  /  x ]. ps  ->  E. x
( ph  /\  ps )
) ) )
2618, 3, 25sylc 58 . . . 4  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  x ]. ps  ->  E. x ( ph  /\ 
ps ) ) )
2726anc2li 542 . . 3  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  x ]. ps  ->  ( E! x ph  /\ 
E. x ( ph  /\ 
ps ) ) ) )
28 eupicka 2208 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  A. x
( ph  ->  ps )
)
2927, 28syl6 31 . 2  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  x ]. ps  ->  A. x ( ph  ->  ps ) ) )
3017, 29impbid 185 1  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  <->  [. ( iota x ph )  /  x ]. ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1528   E.wex 1529    = wceq 1624   [wsb 1631    e. wcel 1685   E!weu 2144   _Vcvv 2789   [.wsbc 2992   iotacio 6250
This theorem is referenced by:  sbaniota  27034
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ral 2549  df-rex 2550  df-v 2791  df-sbc 2993  df-un 3158  df-sn 3647  df-pr 3648  df-uni 3829  df-iota 6252
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