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Theorem iota2 5384
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 2907 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 simpl 444 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  A  e.  _V )
3 simpr 448 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  E! x ph )
4 iota2.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54adantl 453 . . 3  |-  ( ( ( A  e.  _V  /\  E! x ph )  /\  x  =  A
)  ->  ( ph  <->  ps ) )
6 nfv 1626 . . . 4  |-  F/ x  A  e.  _V
7 nfeu1 2248 . . . 4  |-  F/ x E! x ph
86, 7nfan 1836 . . 3  |-  F/ x
( A  e.  _V  /\  E! x ph )
9 nfvd 1627 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  F/ x ps )
10 nfcvd 2524 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  F/_ x A )
112, 3, 5, 8, 9, 10iota2df 5382 . 2  |-  ( ( A  e.  _V  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
121, 11sylan 458 1  |-  ( ( A  e.  B  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E!weu 2238   _Vcvv 2899   iotacio 5356
This theorem is referenced by:  pczpre  13148  pcdiv  13153  unirep  26105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-v 2901  df-sbc 3105  df-un 3268  df-sn 3763  df-pr 3764  df-uni 3958  df-iota 5358
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