MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iota2 Unicode version

Theorem iota2 5261
Description: The unique element such that  ph. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
Hypothesis
Ref Expression
iota2.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
iota2  |-  ( ( A  e.  B  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem iota2
StepHypRef Expression
1 elex 2809 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 simpl 443 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  A  e.  _V )
3 simpr 447 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  E! x ph )
4 iota2.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54adantl 452 . . 3  |-  ( ( ( A  e.  _V  /\  E! x ph )  /\  x  =  A
)  ->  ( ph  <->  ps ) )
6 nfv 1609 . . . 4  |-  F/ x  A  e.  _V
7 nfeu1 2166 . . . 4  |-  F/ x E! x ph
86, 7nfan 1783 . . 3  |-  F/ x
( A  e.  _V  /\  E! x ph )
9 nfvd 1610 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  F/ x ps )
10 nfcvd 2433 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  F/_ x A )
112, 3, 5, 8, 9, 10iota2df 5259 . 2  |-  ( ( A  e.  _V  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
121, 11sylan 457 1  |-  ( ( A  e.  B  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E!weu 2156   _Vcvv 2801   iotacio 5233
This theorem is referenced by:  pczpre  12916  pcdiv  12921  euuni2OLD  26451  unirep  26452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005  df-un 3170  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235
  Copyright terms: Public domain W3C validator