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Theorem sbiota1 27624
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 2287 . . . 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 5439 . . 3  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
4 iotaval 5432 . . . . . 6  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
54eqcomd 2443 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  y  =  ( iota x ph ) )
6 spsbim 2136 . . . . . . . 8  |-  ( A. x ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) )
7 sbsbc 3167 . . . . . . . 8  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
8 sbsbc 3167 . . . . . . . 8  |-  ( [ y  /  x ] ps 
<-> 
[. y  /  x ]. ps )
96, 7, 83imtr3g 262 . . . . . . 7  |-  ( A. x ( ph  ->  ps )  ->  ( [. y  /  x ]. ph  ->  [. y  /  x ]. ps ) )
10 dfsbcq 3165 . . . . . . . 8  |-  ( y  =  ( iota x ph )  ->  ( [. y  /  x ]. ph  <->  [. ( iota
x ph )  /  x ]. ph ) )
11 dfsbcq 3165 . . . . . . . 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 75 . . . . 5  |-  ( y  =  ( iota x ph )  ->  ( [. ( iota x ph )  /  x ]. ph  ->  ( A. x ( ph  ->  ps )  ->  [. ( iota x ph )  /  x ]. ps ) ) )
155, 14syl 16 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( [. ( iota x ph )  /  x ]. ph  ->  ( A. x ( ph  ->  ps )  ->  [. ( iota x ph )  /  x ]. ps ) ) )
1615exlimiv 1645 . . 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 59 . 2  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  ->  [. ( iota
x ph )  /  x ]. ps ) )
18 iotaexeu 27608 . . . . 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 3205 . . . . . . . 8  |-  ( [. y  /  x ]. ( ph  /\  ps )  <->  ( [. y  /  x ]. ph  /\  [. y  /  x ]. ps ) )
22 spesbc 3244 . . . . . . . 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 3013 . . . . . 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 59 . . . 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 2347 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  A. x
( ph  ->  ps )
)
2927, 28syl6 32 . 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 4    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653   [wsb 1659    e. wcel 1726   E!weu 2283   _Vcvv 2958   [.wsbc 3163   iotacio 5419
This theorem is referenced by:  sbaniota  27625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-sbc 3164  df-un 3327  df-sn 3822  df-pr 3823  df-uni 4018  df-iota 5421
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