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Theorem iota2d 5108
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 1508 . 2  |-  F/ x ph
5 nfvd 1509 . 2  |-  ( ph  ->  F/ x ch )
6 nfcvd 2280 . 2  |-  ( ph  -> 
F/_ x B )
71, 2, 3, 4, 5, 6iota2df 5107 1  |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   E!weu 1997   iotacio 5081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-sn 3528  df-pr 3529  df-uni 3732  df-iota 5083
This theorem is referenced by: (None)
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