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Theorem sbiota1 36437
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 2270 . . . 4  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
21biimpi 197 . . 3  |-  ( E! x ph  ->  E. y A. x ( ph  <->  x  =  y ) )
3 iota4 5583 . . 3  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
4 iotaval 5576 . . . . . 6  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
54eqcomd 2437 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
6 spsbim 2189 . . . . . . . 8  |-  ( A. x ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) )
7 sbsbc 3309 . . . . . . . 8  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
8 sbsbc 3309 . . . . . . . 8  |-  ( [ y  /  x ] ps 
<-> 
[. y  /  x ]. ps )
96, 7, 83imtr3g 272 . . . . . . 7  |-  ( A. x ( ph  ->  ps )  ->  ( [. y  /  x ]. ph  ->  [. y  /  x ]. ps ) )
10 dfsbcq 3307 . . . . . . . 8  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  x ]. ph  <->  [. ( iota
x ph )  /  x ]. ph ) )
11 dfsbcq 3307 . . . . . . . 8  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  x ]. ps  <->  [. ( iota x ph )  /  x ]. ps ) )
1210, 11imbi12d 321 . . . . . . 7  |-  ( y  =  ( iota x ph )  ->  ( (
[. y  /  x ]. ph  ->  [. y  /  x ]. ps )  <->  ( [. ( iota x ph )  /  x ]. ph  ->  [. ( iota x ph )  /  x ]. ps ) ) )
139, 12syl5ib 222 . . . . . 6  |-  ( y  =  ( iota x ph )  ->  ( A. x ( ph  ->  ps )  ->  ( [. ( iota x ph )  /  x ]. ph  ->  [. ( iota x ph )  /  x ]. ps ) ) )
1413com23 81 . . . . 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 1769 . . 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 62 . 2  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  ->  [. ( iota
x ph )  /  x ]. ps ) )
18 iotaexeu 36421 . . . . 5  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
1910, 11anbi12d 715 . . . . . . . 8  |-  ( y  =  ( iota x ph )  ->  ( (
[. y  /  x ]. ph  /\  [. y  /  x ]. ps )  <->  (
[. ( iota x ph )  /  x ]. ph  /\  [. ( iota x ph )  /  x ]. ps ) ) )
2019imbi1d 318 . . . . . . 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 3348 . . . . . . . 8  |-  ( [. y  /  x ]. ( ph  /\  ps )  <->  ( [. y  /  x ]. ph  /\  [. y  /  x ]. ps ) )
22 spesbc 3387 . . . . . . . 8  |-  ( [. y  /  x ]. ( ph  /\  ps )  ->  E. x ( ph  /\  ps ) )
2321, 22sylbir 216 . . . . . . 7  |-  ( (
[. y  /  x ]. ph  /\  [. y  /  x ]. ps )  ->  E. x ( ph  /\ 
ps ) )
2420, 23vtoclg 3145 . . . . . 6  |-  ( ( iota x ph )  e.  _V  ->  ( ( [. ( iota x ph )  /  x ]. ph  /\  [. ( iota x ph )  /  x ]. ps )  ->  E. x ( ph  /\ 
ps ) ) )
2524expd 437 . . . . 5  |-  ( ( iota x ph )  e.  _V  ->  ( [. ( iota x ph )  /  x ]. ph  ->  (
[. ( iota x ph )  /  x ]. ps  ->  E. x
( ph  /\  ps )
) ) )
2618, 3, 25sylc 62 . . . 4  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  x ]. ps  ->  E. x ( ph  /\ 
ps ) ) )
2726anc2li 559 . . 3  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  x ]. ps  ->  ( E! x ph  /\ 
E. x ( ph  /\ 
ps ) ) ) )
28 eupicka 2337 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  A. x
( ph  ->  ps )
)
2927, 28syl6 34 . 2  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  x ]. ps  ->  A. x ( ph  ->  ps ) ) )
3017, 29impbid 193 1  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  <->  [. ( iota x ph )  /  x ]. ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1659   [wsb 1789    e. wcel 1870   E!weu 2266   _Vcvv 3087   [.wsbc 3305   iotacio 5563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-v 3089  df-sbc 3306  df-un 3447  df-sn 4003  df-pr 4005  df-uni 4223  df-iota 5565
This theorem is referenced by:  sbaniota  36438
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