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Theorem riota2f 5865
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 2340 . 2  |-  F/ x  B  e.  A
31a1i 9 . 2  |-  ( B  e.  A  ->  F/_ x B )
4 riota2f.2 . . 3  |-  F/ x ps
54a1i 9 . 2  |-  ( B  e.  A  ->  F/ x ps )
6 id 19 . 2  |-  ( B  e.  A  ->  B  e.  A )
7 riota2f.3 . . 3  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
87adantl 277 . 2  |-  ( ( B  e.  A  /\  x  =  B )  ->  ( ph  <->  ps )
)
92, 3, 5, 6, 8riota2df 5864 1  |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  (
iota_ x  e.  A  ph )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1363   F/wnf 1470    e. wcel 2158   F/_wnfc 2316   E!wreu 2467   iota_crio 5843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-reu 2472  df-v 2751  df-sbc 2975  df-un 3145  df-sn 3610  df-pr 3611  df-uni 3822  df-iota 5190  df-riota 5844
This theorem is referenced by:  riota2  5866  riotaprop  5867  riotass2  5870  riotass  5871
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