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Theorem riotasbc 7271
Description: Substitution law for descriptions. Compare iotasbc 42061. (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 4021 . . 3 {𝑥𝐴𝜑} ⊆ {𝑥𝜑}
2 riotacl2 7269 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sselid 3921 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝜑})
4 df-sbc 3719 . 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 2101  {cab 2710  ∃!wreu 3219  {crab 3221  [wsbc 3718  crio 7251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-12 2166  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-un 3894  df-in 3896  df-ss 3906  df-sn 4565  df-pr 4567  df-uni 4842  df-iota 6399  df-riota 7252
This theorem is referenced by:  riotass2  7283  riotass  7284  cjth  14842  joinlem  18129  meetlem  18143  finxpreclem4  35593  poimirlem26  35831  riotasvd  36996  lshpkrlem3  37152
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