MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riotasbc Structured version   Visualization version   GIF version

Theorem riotasbc 7333
Description: Substitution law for descriptions. Compare iotasbc 42706. (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 4044 . . 3 {𝑥𝐴𝜑} ⊆ {𝑥𝜑}
2 riotacl2 7331 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sselid 3943 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝜑})
4 df-sbc 3741 . 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 2107  {cab 2714  ∃!wreu 3352  {crab 3408  [wsbc 3740  crio 7313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-un 3916  df-in 3918  df-ss 3928  df-sn 4588  df-pr 4590  df-uni 4867  df-iota 6449  df-riota 7314
This theorem is referenced by:  riotass2  7345  riotass  7346  cjth  14989  joinlem  18273  meetlem  18287  finxpreclem4  35868  poimirlem26  36107  riotasvd  37421  lshpkrlem3  37577
  Copyright terms: Public domain W3C validator