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Theorem iota2df 5434
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 448 . . . 4  |-  ( (
ph  /\  x  =  B )  ->  x  =  B )
43eqeq2d 2446 . . 3  |-  ( (
ph  /\  x  =  B )  ->  (
( iota x ps )  =  x  <->  ( iota x ps )  =  B
) )
52, 4bibi12d 313 . 2  |-  ( (
ph  /\  x  =  B )  ->  (
( ps  <->  ( iota x ps )  =  x )  <->  ( ch  <->  ( iota x ps )  =  B ) ) )
6 iota2df.2 . . 3  |-  ( ph  ->  E! x ps )
7 iota1 5424 . . 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 5412 . . . . 5  |-  F/_ x
( iota x ps )
1312a1i 11 . . . 4  |-  ( ph  -> 
F/_ x ( iota
x ps ) )
1413, 10nfeqd 2585 . . 3  |-  ( ph  ->  F/ x ( iota
x ps )  =  B )
1511, 14nfbid 1854 . 2  |-  ( ph  ->  F/ x ( ch  <->  ( iota x ps )  =  B ) )
161, 5, 8, 9, 10, 15vtocldf 2995 1  |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   E!weu 2280   F/_wnfc 2558   iotacio 5408
This theorem is referenced by:  iota2d  5435  iota2  5436  opiota  6527  riota2df  6562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-sbc 3154  df-un 3317  df-sn 3812  df-pr 3813  df-uni 4008  df-iota 5410
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