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Theorem riotasbc 7391
Description: Substitution law for descriptions. Compare iotasbc 44130. (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 4079 . . 3 {𝑥𝐴𝜑} ⊆ {𝑥𝜑}
2 riotacl2 7389 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sselid 3976 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝜑})
4 df-sbc 3776 . 2 ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐴 𝜑) ∈ {𝑥𝜑})
53, 4sylibr 233 1 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  {cab 2703  ∃!wreu 3362  {crab 3419  [wsbc 3775  crio 7371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-un 3951  df-ss 3963  df-sn 4624  df-pr 4626  df-uni 4906  df-iota 6498  df-riota 7372
This theorem is referenced by:  riotass2  7403  riotass  7404  cjth  15103  joinlem  18403  meetlem  18417  finxpreclem4  37114  poimirlem26  37360  riotasvd  38667  lshpkrlem3  38823  tfsconcatfv  43044
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