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Theorem riotasbc 6854
Description: Substitution law for descriptions. Compare iotasbc 39401. (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 3887 . . 3 {𝑥𝐴𝜑} ⊆ {𝑥𝜑}
2 riotacl2 6852 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sseldi 3796 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝜑})
4 df-sbc 3634 . 2 ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐴 𝜑) ∈ {𝑥𝜑})
53, 4sylibr 226 1 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  {cab 2785  ∃!wreu 3091  {crab 3093  [wsbc 3633  crio 6838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-un 3774  df-in 3776  df-ss 3783  df-sn 4369  df-pr 4371  df-uni 4629  df-iota 6064  df-riota 6839
This theorem is referenced by:  riotass2  6866  riotass  6867  cjth  14184  joinlem  17326  meetlem  17340  finxpreclem4  33729  poimirlem26  33924  riotasvd  34977  lshpkrlem3  35133
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