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Theorem iota2d 5244
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 1605 . 2  |-  F/ x ph
5 nfvd 1606 . 2  |-  ( ph  ->  F/ x ch )
6 nfcvd 2420 . 2  |-  ( ph  -> 
F/_ x B )
71, 2, 3, 4, 5, 6iota2df 5243 1  |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E!weu 2143   iotacio 5217
This theorem is referenced by:  erov  6755  q1peqb  19540  psgnvalii  27432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-un 3157  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219
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