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Theorem iota2d 5222
Description: A condition that allows us to represent "the unique element such that  ph " with a class expression  A. (Contributed by NM, 30-Dec-2014.)
Hypotheses
Ref Expression
iota2df.1  |-  ( ph  ->  B  e.  V )
iota2df.2  |-  ( ph  ->  E! x ps )
iota2df.3  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
iota2d  |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )
Distinct variable groups:    x, B    ph, x    ch, x
Allowed substitution hints:    ps( x)    V( x)

Proof of Theorem iota2d
StepHypRef Expression
1 iota2df.1 . 2  |-  ( ph  ->  B  e.  V )
2 iota2df.2 . 2  |-  ( ph  ->  E! x ps )
3 iota2df.3 . 2  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
4 nfv 1539 . 2  |-  F/ x ph
5 nfvd 1540 . 2  |-  ( ph  ->  F/ x ch )
6 nfcvd 2333 . 2  |-  ( ph  -> 
F/_ x B )
71, 2, 3, 4, 5, 6iota2df 5221 1  |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   E!weu 2038    e. wcel 2160   iotacio 5194
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 3613  df-pr 3614  df-uni 3825  df-iota 5196
This theorem is referenced by: (None)
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