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Theorem riotacl 5374
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 3003 . 2 {x Aφ} ⊆ A
2 riotacl2 5373 . 2 (∃!x A φ → (x A φ) {x Aφ})
31, 2sseldi 2921 1 (∃!x A φ → (x A φ) A)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1375  ∃!wreu 2284  {crab 2286  crio 5359
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2288  df-reu 2289  df-rab 2291  df-v 2535  df-sbc 2740  df-un 2900  df-in 2902  df-ss 2909  df-sn 3333  df-pr 3334  df-uni 3533  df-iota 4761  df-riota 5360
This theorem is referenced by:  riotaprop  5383  riotass2  5386  riotass  5387  acexmidlemcase  5400
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