MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iota4an Unicode version

Theorem iota4an 5238
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 5237 . 2  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ( ph  /\  ps ) )
2 iotaex 5236 . . . 4  |-  ( iota
x ( ph  /\  ps ) )  e.  _V
3 simpl 443 . . . . 5  |-  ( (
ph  /\  ps )  ->  ph )
43sbcth 3005 . . . 4  |-  ( ( iota x ( ph  /\ 
ps ) )  e. 
_V  ->  [. ( iota x
( ph  /\  ps )
)  /  x ]. ( ( ph  /\  ps )  ->  ph )
)
52, 4ax-mp 8 . . 3  |-  [. ( iota x ( ph  /\  ps ) )  /  x ]. ( ( ph  /\  ps )  ->  ph )
6 sbcimg 3032 . . . 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 8 . . 3  |-  ( [. ( iota x ( ph  /\ 
ps ) )  /  x ]. ( ( ph  /\ 
ps )  ->  ph )  <->  (
[. ( iota x
( ph  /\  ps )
)  /  x ]. ( ph  /\  ps )  ->  [. ( iota x
( ph  /\  ps )
)  /  x ]. ph ) )
85, 7mpbi 199 . 2  |-  ( [. ( iota x ( ph  /\ 
ps ) )  /  x ]. ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ph )
91, 8syl 15 1  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   E!weu 2143   _Vcvv 2788   [.wsbc 2991   iotacio 5217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219
  Copyright terms: Public domain W3C validator