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Theorem iota2df 5004
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 108 . . . 4  |-  ( (
ph  /\  x  =  B )  ->  x  =  B )
43eqeq2d 2099 . . 3  |-  ( (
ph  /\  x  =  B )  ->  (
( iota x ps )  =  x  <->  ( iota x ps )  =  B
) )
52, 4bibi12d 233 . 2  |-  ( (
ph  /\  x  =  B )  ->  (
( ps  <->  ( iota x ps )  =  x )  <->  ( ch  <->  ( iota x ps )  =  B ) ) )
6 iota2df.2 . . 3  |-  ( ph  ->  E! x ps )
7 iota1 4994 . . 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 4982 . . . . 5  |-  F/_ x
( iota x ps )
1312a1i 9 . . . 4  |-  ( ph  -> 
F/_ x ( iota
x ps ) )
1413, 10nfeqd 2243 . . 3  |-  ( ph  ->  F/ x ( iota
x ps )  =  B )
1511, 14nfbid 1525 . 2  |-  ( ph  ->  F/ x ( ch  <->  ( iota x ps )  =  B ) )
161, 5, 8, 9, 10, 15vtocldf 2670 1  |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   F/wnf 1394    e. wcel 1438   E!weu 1948   F/_wnfc 2215   iotacio 4978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-sn 3452  df-pr 3453  df-uni 3654  df-iota 4980
This theorem is referenced by:  iota2d  5005  iota2  5006  riota2df  5628
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