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Theorem riotasbc 6320
Description: Substitution law for descriptions. Compare iotasbc 27619. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
riotasbc  |-  ( E! x  e.  A  ph  ->  [. ( iota_ x  e.  A ph )  /  x ]. ph )

Proof of Theorem riotasbc
StepHypRef Expression
1 rabssab 3259 . . 3  |-  { x  e.  A  |  ph }  C_ 
{ x  |  ph }
2 riotacl2 6318 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  { x  e.  A  |  ph }
)
31, 2sseldi 3178 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  { x  |  ph } )
4 df-sbc 2992 . 2  |-  ( [. ( iota_ x  e.  A ph )  /  x ]. ph  <->  ( iota_ x  e.  A ph )  e. 
{ x  |  ph } )
53, 4sylibr 203 1  |-  ( E! x  e.  A  ph  ->  [. ( iota_ x  e.  A ph )  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   {cab 2269   E!wreu 2545   {crab 2547   [.wsbc 2991   iota_crio 6297
This theorem is referenced by:  riotass2  6332  riotass  6333  riotasvd  6347  cjth  11588  joinlem  14124  meetlem  14131  lshpkrlem3  29302
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-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-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-un 3157  df-in 3159  df-ss 3166  df-if 3566  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219  df-riota 6304
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