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

Theorem riotasbc 6789
Description: Substitution law for descriptions. Compare iotasbc 39122. (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 3832 . . 3 {𝑥𝐴𝜑} ⊆ {𝑥𝜑}
2 riotacl2 6787 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sseldi 3742 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝜑})
4 df-sbc 3577 . 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 2139  {cab 2746  ∃!wreu 3052  {crab 3054  [wsbc 3576  crio 6773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-un 3720  df-in 3722  df-ss 3729  df-sn 4322  df-pr 4324  df-uni 4589  df-iota 6012  df-riota 6774
This theorem is referenced by:  riotass2  6801  riotass  6802  cjth  14042  joinlem  17212  meetlem  17226  finxpreclem4  33542  poimirlem26  33748  riotasvd  34745  lshpkrlem3  34902
  Copyright terms: Public domain W3C validator