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Theorem riotasbc 6336
Description: Substitution law for descriptions. Compare iotasbc 27722. (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 3272 . . 3  |-  { x  e.  A  |  ph }  C_ 
{ x  |  ph }
2 riotacl2 6334 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  { x  e.  A  |  ph }
)
31, 2sseldi 3191 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  { x  |  ph } )
4 df-sbc 3005 . 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 1696   {cab 2282   E!wreu 2558   {crab 2560   [.wsbc 3004   iota_crio 6313
This theorem is referenced by:  riotass2  6348  riotass  6349  riotasvd  6363  cjth  11604  joinlem  14140  meetlem  14147  lshpkrlem3  29924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-un 3170  df-in 3172  df-ss 3179  df-if 3579  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235  df-riota 6320
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