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Theorem riota2f 6264
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 2621 . 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 6263 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 1383   F/wnf 1603    e. wcel 1804   F/_wnfc 2591   E!wreu 2795   iota_crio 6241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-reu 2800  df-v 3097  df-sbc 3314  df-un 3466  df-sn 4015  df-pr 4017  df-uni 4235  df-iota 5541  df-riota 6242
This theorem is referenced by:  riota2  6265  riotaprop  6266  riotass2  6269  riotass  6270  riotaxfrd  6273  cdlemksv2  36448  cdlemkuv2  36468  cdlemk36  36514
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