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Theorem riota2f 6074
Description: This theorem shows a condition that allows us to represent a descriptor with a class expression  B. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riota2f.1  |-  F/_ x B
riota2f.2  |-  F/ x ps
riota2f.3  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
riota2f  |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  (
iota_ x  e.  A  ph )  =  B ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    B( x)

Proof of Theorem riota2f
StepHypRef Expression
1 riota2f.1 . . 3  |-  F/_ x B
21nfel1 2589 . 2  |-  F/ x  B  e.  A
31a1i 11 . 2  |-  ( B  e.  A  ->  F/_ x B )
4 riota2f.2 . . 3  |-  F/ x ps
54a1i 11 . 2  |-  ( B  e.  A  ->  F/ x ps )
6 id 22 . 2  |-  ( B  e.  A  ->  B  e.  A )
7 riota2f.3 . . 3  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
87adantl 466 . 2  |-  ( ( B  e.  A  /\  x  =  B )  ->  ( ph  <->  ps )
)
92, 3, 5, 6, 8riota2df 6073 1  |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  (
iota_ x  e.  A  ph )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   F/wnf 1589    e. wcel 1756   F/_wnfc 2566   E!wreu 2717   iota_crio 6051
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 2568  df-ral 2720  df-rex 2721  df-reu 2722  df-v 2974  df-sbc 3187  df-un 3333  df-sn 3878  df-pr 3880  df-uni 4092  df-iota 5381  df-riota 6052
This theorem is referenced by:  riota2  6075  riotaprop  6076  riotass2  6079  riotass  6080  riotaxfrd  6083  cdlemksv2  34491  cdlemkuv2  34511  cdlemk36  34557
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