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Theorem riotasbc 5407
Description: Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
riotasbc (∃!x A φ[(x A φ) / x]φ)

Proof of Theorem riotasbc
StepHypRef Expression
1 rabssab 3004 . . 3 {x Aφ} ⊆ {xφ}
2 riotacl2 5405 . . 3 (∃!x A φ → (x A φ) {x Aφ})
31, 2sseldi 2920 . 2 (∃!x A φ → (x A φ) {xφ})
4 df-sbc 2742 . 2 ([(x A φ) / x]φ ↔ (x A φ) {xφ})
53, 4sylibr 137 1 (∃!x A φ[(x A φ) / x]φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1374  {cab 2008  ∃!wreu 2286  {crab 2288  [wsbc 2741  crio 5392
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 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-uni 3555  df-iota 4794  df-riota 5393
This theorem is referenced by:  riotass2  5418  riotass  5419
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