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Theorem iota2d 5445
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 1630 . 2  |-  F/ x ph
5 nfvd 1631 . 2  |-  ( ph  ->  F/ x ch )
6 nfcvd 2575 . 2  |-  ( ph  -> 
F/_ x B )
71, 2, 3, 4, 5, 6iota2df 5444 1  |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E!weu 2283   iotacio 5418
This theorem is referenced by:  erov  7003  q1peqb  20079  psgnvalii  27411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-sbc 3164  df-un 3327  df-sn 3822  df-pr 3823  df-uni 4018  df-iota 5420
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