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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 3000 . 2 {x Aφ} ⊆ A
2 riotacl2 5403 . 2 (∃!x A φ → (x A φ) {x Aφ})
31, 2sseldi 2918 1 (∃!x A φ → (x A φ) A)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1374  ∃!wreu 2284  {crab 2286  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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  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 2897  df-in 2899  df-ss 2906  df-sn 3354  df-pr 3355  df-uni 3553  df-iota 4792  df-riota 5391
This theorem is referenced by:  riotaprop  5413  riotass2  5416  riotass  5417  acexmidlemcase  5429
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