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Theorem riotasbc 7127
Description: Substitution law for descriptions. Compare iotasbc 40618. (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 4063 . . 3 {𝑥𝐴𝜑} ⊆ {𝑥𝜑}
2 riotacl2 7125 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sseldi 3968 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝜑})
4 df-sbc 3776 . 2 ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐴 𝜑) ∈ {𝑥𝜑})
53, 4sylibr 235 1 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  {cab 2803  ∃!wreu 3144  {crab 3146  [wsbc 3775  crio 7108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-un 3944  df-in 3946  df-ss 3955  df-sn 4564  df-pr 4566  df-uni 4837  df-iota 6311  df-riota 7109
This theorem is referenced by:  riotass2  7139  riotass  7140  cjth  14455  joinlem  17613  meetlem  17627  finxpreclem4  34546  poimirlem26  34787  riotasvd  35961  lshpkrlem3  36117
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