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

Theorem riotasbc 6586
 Description: Substitution law for descriptions. Compare iotasbc 38129. (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 3673 . . 3 {𝑥𝐴𝜑} ⊆ {𝑥𝜑}
2 riotacl2 6584 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sseldi 3585 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝜑})
4 df-sbc 3422 . 2 ([(𝑥𝐴 𝜑) / 𝑥]𝜑 ↔ (𝑥𝐴 𝜑) ∈ {𝑥𝜑})
53, 4sylibr 224 1 (∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1987  {cab 2607  ∃!wreu 2909  {crab 2911  [wsbc 3421  ℩crio 6570 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-un 3564  df-in 3566  df-ss 3573  df-sn 4154  df-pr 4156  df-uni 4408  df-iota 5815  df-riota 6571 This theorem is referenced by:  riotass2  6598  riotass  6599  cjth  13784  joinlem  16939  meetlem  16953  finxpreclem4  32890  poimirlem26  33094  riotasvd  33749  lshpkrlem3  33906
 Copyright terms: Public domain W3C validator