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Theorem iota2df 5417
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 461 . . . 4  |-  ( (
ph  /\  x  =  B )  ->  x  =  B )
43eqeq2d 2454 . . 3  |-  ( (
ph  /\  x  =  B )  ->  (
( iota x ps )  =  x  <->  ( iota x ps )  =  B
) )
52, 4bibi12d 321 . 2  |-  ( (
ph  /\  x  =  B )  ->  (
( ps  <->  ( iota x ps )  =  x )  <->  ( ch  <->  ( iota x ps )  =  B ) ) )
6 iota2df.2 . . 3  |-  ( ph  ->  E! x ps )
7 iota1 5407 . . 3  |-  ( E! x ps  ->  ( ps 
<->  ( iota x ps )  =  x ) )
86, 7syl 16 . 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 5395 . . . . 5  |-  F/_ x
( iota x ps )
1312a1i 11 . . . 4  |-  ( ph  -> 
F/_ x ( iota
x ps ) )
1413, 10nfeqd 2596 . . 3  |-  ( ph  ->  F/ x ( iota
x ps )  =  B )
1511, 14nfbid 1866 . 2  |-  ( ph  ->  F/ x ( ch  <->  ( iota x ps )  =  B ) )
161, 5, 8, 9, 10, 15vtocldf 3033 1  |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   F/wnf 1589    e. wcel 1756   E!weu 2253   F/_wnfc 2575   iotacio 5391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2732  df-rex 2733  df-v 2986  df-sbc 3199  df-un 3345  df-sn 3890  df-pr 3892  df-uni 4104  df-iota 5393
This theorem is referenced by:  iota2d  5418  iota2  5419  riota2df  6085  opiota  6645
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