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Theorem riotasbc 5745
Description: Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
riotasbc (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)

Proof of Theorem riotasbc
StepHypRef Expression
1 rabssab 3184 . . 3 {𝑥𝐴𝜑} ⊆ {𝑥𝜑}
2 riotacl2 5743 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sseldi 3095 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝜑})
4 df-sbc 2910 . 2 ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐴 𝜑) ∈ {𝑥𝜑})
53, 4sylibr 133 1 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1480  {cab 2125  ∃!wreu 2418  {crab 2420  [wsbc 2909  crio 5729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-uni 3737  df-iota 5088  df-riota 5730
This theorem is referenced by:  riotass2  5756  riotass  5757  cjth  10625
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