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Theorem iota2d 5590
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 1755 . 2  |-  F/ x ph
5 nfvd 1756 . 2  |-  ( ph  ->  F/ x ch )
6 nfcvd 2581 . 2  |-  ( ph  -> 
F/_ x B )
71, 2, 3, 4, 5, 6iota2df 5589 1  |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   E!weu 2269   iotacio 5563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082  df-sbc 3300  df-un 3441  df-sn 3999  df-pr 4001  df-uni 4220  df-iota 5565
This theorem is referenced by:  erov  7472  psgnvalii  17150  q1peqb  23104
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