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Theorem iota4an 5553
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 5552 . 2  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ( ph  /\  ps ) )
2 iotaex 5551 . . . 4  |-  ( iota
x ( ph  /\  ps ) )  e.  _V
3 simpl 455 . . . . 5  |-  ( (
ph  /\  ps )  ->  ph )
43sbcth 3339 . . . 4  |-  ( ( iota x ( ph  /\ 
ps ) )  e. 
_V  ->  [. ( iota x
( ph  /\  ps )
)  /  x ]. ( ( ph  /\  ps )  ->  ph )
)
52, 4ax-mp 5 . . 3  |-  [. ( iota x ( ph  /\  ps ) )  /  x ]. ( ( ph  /\  ps )  ->  ph )
6 sbcimg 3366 . . . 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 5 . . 3  |-  ( [. ( iota x ( ph  /\ 
ps ) )  /  x ]. ( ( ph  /\ 
ps )  ->  ph )  <->  (
[. ( iota x
( ph  /\  ps )
)  /  x ]. ( ph  /\  ps )  ->  [. ( iota x
( ph  /\  ps )
)  /  x ]. ph ) )
85, 7mpbi 208 . 2  |-  ( [. ( iota x ( ph  /\ 
ps ) )  /  x ]. ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ph )
91, 8syl 16 1  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1823   E!weu 2284   _Vcvv 3106   [.wsbc 3324   iotacio 5532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019  df-uni 4236  df-iota 5534
This theorem is referenced by: (None)
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