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Theorem riota2f 6326
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 2429 . 2  |-  F/ x  B  e.  A
31a1i 10 . 2  |-  ( B  e.  A  ->  F/_ x B )
4 riota2f.2 . . 3  |-  F/ x ps
54a1i 10 . 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 452 . 2  |-  ( ( B  e.  A  /\  x  =  B )  ->  ( ph  <->  ps )
)
92, 3, 5, 6, 8riota2df 6325 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    <-> wb 176    /\ wa 358   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406   E!wreu 2545   iota_crio 6297
This theorem is referenced by:  riota2  6327  riotaprop  6328  riotass2  6332  riotass  6333  riotaxfrd  6336  cvmliftphtlem  23848  cdlemksv2  31036  cdlemkuv2  31056
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-reu 2550  df-v 2790  df-sbc 2992  df-un 3157  df-if 3566  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219  df-riota 6304
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