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Theorem iota2 5228
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 2763 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 simpl 109 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  A  e.  _V )
3 simpr 110 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  E! x ph )
4 iota2.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54adantl 277 . . 3  |-  ( ( ( A  e.  _V  /\  E! x ph )  /\  x  =  A
)  ->  ( ph  <->  ps ) )
6 nfv 1539 . . . 4  |-  F/ x  A  e.  _V
7 nfeu1 2049 . . . 4  |-  F/ x E! x ph
86, 7nfan 1576 . . 3  |-  F/ x
( A  e.  _V  /\  E! x ph )
9 nfvd 1540 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  F/ x ps )
10 nfcvd 2333 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  F/_ x A )
112, 3, 5, 8, 9, 10iota2df 5224 . 2  |-  ( ( A  e.  _V  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
121, 11sylan 283 1  |-  ( ( A  e.  B  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   E!weu 2038    e. wcel 2160   _Vcvv 2752   iotacio 5197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-sn 3616  df-pr 3617  df-uni 3828  df-iota 5199
This theorem is referenced by:  iotam  5230  pczpre  12340  pcdiv  12345  gsum0g  12882  gsumval2  12883
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