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Theorem sbiota1 27737
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 ) )

Proof of Theorem sbiota1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-eu 2160 . . . 4  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
21biimpi 186 . . 3  |-  ( E! x ph  ->  E. y A. x ( ph  <->  x  =  y ) )
3 iota4 5253 . . 3  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
4 iotaval 5246 . . . . . 6  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
54eqcomd 2301 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
6 spsbim 2029 . . . . . . . 8  |-  ( A. x ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) )
7 sbsbc 3008 . . . . . . . 8  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
8 sbsbc 3008 . . . . . . . 8  |-  ( [ y  /  x ] ps 
<-> 
[. y  /  x ]. ps )
96, 7, 83imtr3g 260 . . . . . . 7  |-  ( A. x ( ph  ->  ps )  ->  ( [. y  /  x ]. ph  ->  [. y  /  x ]. ps ) )
10 dfsbcq 3006 . . . . . . . 8  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  x ]. ph  <->  [. ( iota
x ph )  /  x ]. ph ) )
11 dfsbcq 3006 . . . . . . . 8  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  x ]. ps  <->  [. ( iota x ph )  /  x ]. ps ) )
1210, 11imbi12d 311 . . . . . . 7  |-  ( y  =  ( iota x ph )  ->  ( (
[. y  /  x ]. ph  ->  [. y  /  x ]. ps )  <->  ( [. ( iota x ph )  /  x ]. ph  ->  [. ( iota x ph )  /  x ]. ps ) ) )
139, 12syl5ib 210 . . . . . 6  |-  ( y  =  ( iota x ph )  ->  ( A. x ( ph  ->  ps )  ->  ( [. ( iota x ph )  /  x ]. ph  ->  [. ( iota x ph )  /  x ]. ps ) ) )
1413com23 72 . . . . 5  |-  ( y  =  ( iota x ph )  ->  ( [. ( iota x ph )  /  x ]. ph  ->  ( A. x ( ph  ->  ps )  ->  [. ( iota x ph )  /  x ]. ps ) ) )
155, 14syl 15 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( [. ( iota x ph )  /  x ]. ph  ->  ( A. x ( ph  ->  ps )  ->  [. ( iota x ph )  /  x ]. ps ) ) )
1615exlimiv 1624 . . 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 56 . 2  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  ->  [. ( iota
x ph )  /  x ]. ps ) )
18 iotaexeu 27721 . . . . 5  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
1910, 11anbi12d 691 . . . . . . . 8  |-  ( y  =  ( iota x ph )  ->  ( (
[. y  /  x ]. ph  /\  [. y  /  x ]. ps )  <->  (
[. ( iota x ph )  /  x ]. ph  /\  [. ( iota x ph )  /  x ]. ps ) ) )
2019imbi1d 308 . . . . . . 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 3046 . . . . . . . 8  |-  ( [. y  /  x ]. ( ph  /\  ps )  <->  ( [. y  /  x ]. ph  /\  [. y  /  x ]. ps ) )
22 spesbc 3085 . . . . . . . 8  |-  ( [. y  /  x ]. ( ph  /\  ps )  ->  E. x ( ph  /\  ps ) )
2321, 22sylbir 204 . . . . . . 7  |-  ( (
[. y  /  x ]. ph  /\  [. y  /  x ]. ps )  ->  E. x ( ph  /\ 
ps ) )
2420, 23vtoclg 2856 . . . . . 6  |-  ( ( iota x ph )  e.  _V  ->  ( ( [. ( iota x ph )  /  x ]. ph  /\  [. ( iota x ph )  /  x ]. ps )  ->  E. x ( ph  /\ 
ps ) ) )
2524exp3a 425 . . . . 5  |-  ( ( iota x ph )  e.  _V  ->  ( [. ( iota x ph )  /  x ]. ph  ->  (
[. ( iota x ph )  /  x ]. ps  ->  E. x
( ph  /\  ps )
) ) )
2618, 3, 25sylc 56 . . . 4  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  x ]. ps  ->  E. x ( ph  /\ 
ps ) ) )
2726anc2li 540 . . 3  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  x ]. ps  ->  ( E! x ph  /\ 
E. x ( ph  /\ 
ps ) ) ) )
28 eupicka 2220 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  A. x
( ph  ->  ps )
)
2927, 28syl6 29 . 2  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  x ]. ps  ->  A. x ( ph  ->  ps ) ) )
3017, 29impbid 183 1  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  <->  [. ( iota x ph )  /  x ]. ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632   [wsb 1638    e. wcel 1696   E!weu 2156   _Vcvv 2801   [.wsbc 3004   iotacio 5233
This theorem is referenced by:  sbaniota  27738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005  df-un 3170  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235
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