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Theorem riotacl 5404
Description: Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
Assertion
Ref Expression
riotacl (∃!x A φ → (x A φ) A)
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem riotacl
StepHypRef Expression
1 ssrab2 3001 . 2 {x Aφ} ⊆ A
2 riotacl2 5403 . 2 (∃!x A φ → (x A φ) {x Aφ})
31, 2sseldi 2919 1 (∃!x A φ → (x A φ) A)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1375  ∃!wreu 2285  {crab 2287  crio 5390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-rex 2289  df-reu 2290  df-rab 2292  df-v 2536  df-sbc 2741  df-un 2898  df-in 2900  df-ss 2907  df-sn 3355  df-pr 3356  df-uni 3554  df-iota 4792  df-riota 5391
This theorem is referenced by:  riotaprop  5413  riotass2  5416  riotass  5417  acexmidlemcase  5429
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