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Theorem iota4an 6271
Description: Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
iota4an  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ph )

Proof of Theorem iota4an
StepHypRef Expression
1 iota4 6270 . 2  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ( ph  /\  ps ) )
2 iotaex 6269 . . . 4  |-  ( iota
x ( ph  /\  ps ) )  e.  _V
3 simpl 445 . . . . 5  |-  ( (
ph  /\  ps )  ->  ph )
43sbcth 3006 . . . 4  |-  ( ( iota x ( ph  /\ 
ps ) )  e. 
_V  ->  [. ( iota x
( ph  /\  ps )
)  /  x ]. ( ( ph  /\  ps )  ->  ph )
)
52, 4ax-mp 10 . . 3  |-  [. ( iota x ( ph  /\  ps ) )  /  x ]. ( ( ph  /\  ps )  ->  ph )
6 sbcimg 3033 . . . 4  |-  ( ( iota x ( ph  /\ 
ps ) )  e. 
_V  ->  ( [. ( iota x ( ph  /\  ps ) )  /  x ]. ( ( ph  /\  ps )  ->  ph )  <->  (
[. ( iota x
( ph  /\  ps )
)  /  x ]. ( ph  /\  ps )  ->  [. ( iota x
( ph  /\  ps )
)  /  x ]. ph ) ) )
72, 6ax-mp 10 . . 3  |-  ( [. ( iota x ( ph  /\ 
ps ) )  /  x ]. ( ( ph  /\ 
ps )  ->  ph )  <->  (
[. ( iota x
( ph  /\  ps )
)  /  x ]. ( ph  /\  ps )  ->  [. ( iota x
( ph  /\  ps )
)  /  x ]. ph ) )
85, 7mpbi 201 . 2  |-  ( [. ( iota x ( ph  /\ 
ps ) )  /  x ]. ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ph )
91, 8syl 17 1  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1688   E!weu 2144   _Vcvv 2789   [.wsbc 2992   iotacio 6250
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-nul 4150
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-sn 3647  df-pr 3648  df-uni 3829  df-iota 6252
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