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Theorem riotasbc 6565
Description: Substitution law for descriptions. Compare iotasbc 27596. (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 3430 . . 3  |-  { x  e.  A  |  ph }  C_ 
{ x  |  ph }
2 riotacl2 6563 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  { x  e.  A  |  ph }
)
31, 2sseldi 3346 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  { x  |  ph } )
4 df-sbc 3162 . 2  |-  ( [. ( iota_ x  e.  A ph )  /  x ]. ph  <->  ( iota_ x  e.  A ph )  e. 
{ x  |  ph } )
53, 4sylibr 204 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 1725   {cab 2422   E!wreu 2707   {crab 2709   [.wsbc 3161   iota_crio 6542
This theorem is referenced by:  riotass2  6577  riotass  6578  riotasvd  6592  cjth  11908  joinlem  14447  meetlem  14454  lshpkrlem3  29910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-un 3325  df-in 3327  df-ss 3334  df-if 3740  df-sn 3820  df-pr 3821  df-uni 4016  df-iota 5418  df-riota 6549
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