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Theorem iota2df 5107
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 ) )
iota2df.4  |-  F/ x ph
iota2df.5  |-  ( ph  ->  F/ x ch )
iota2df.6  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
iota2df  |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )

Proof of Theorem iota2df
StepHypRef Expression
1 iota2df.1 . 2  |-  ( ph  ->  B  e.  V )
2 iota2df.3 . . 3  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
3 simpr 109 . . . 4  |-  ( (
ph  /\  x  =  B )  ->  x  =  B )
43eqeq2d 2149 . . 3  |-  ( (
ph  /\  x  =  B )  ->  (
( iota x ps )  =  x  <->  ( iota x ps )  =  B
) )
52, 4bibi12d 234 . 2  |-  ( (
ph  /\  x  =  B )  ->  (
( ps  <->  ( iota x ps )  =  x )  <->  ( ch  <->  ( iota x ps )  =  B ) ) )
6 iota2df.2 . . 3  |-  ( ph  ->  E! x ps )
7 iota1 5097 . . 3  |-  ( E! x ps  ->  ( ps 
<->  ( iota x ps )  =  x ) )
86, 7syl 14 . 2  |-  ( ph  ->  ( ps  <->  ( iota x ps )  =  x ) )
9 iota2df.4 . 2  |-  F/ x ph
10 iota2df.6 . 2  |-  ( ph  -> 
F/_ x B )
11 iota2df.5 . . 3  |-  ( ph  ->  F/ x ch )
12 nfiota1 5085 . . . . 5  |-  F/_ x
( iota x ps )
1312a1i 9 . . . 4  |-  ( ph  -> 
F/_ x ( iota
x ps ) )
1413, 10nfeqd 2294 . . 3  |-  ( ph  ->  F/ x ( iota
x ps )  =  B )
1511, 14nfbid 1567 . 2  |-  ( ph  ->  F/ x ( ch  <->  ( iota x ps )  =  B ) )
161, 5, 8, 9, 10, 15vtocldf 2732 1  |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   F/wnf 1436    e. wcel 1480   E!weu 1997   F/_wnfc 2266   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:  iota2d  5108  iota2  5109  riota2df  5743
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