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

Theorem ssrd 3923
 Description: Deduction based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
Hypotheses
Ref Expression
ssrd.0 𝑥𝜑
ssrd.1 𝑥𝐴
ssrd.2 𝑥𝐵
ssrd.3 (𝜑 → (𝑥𝐴𝑥𝐵))
Assertion
Ref Expression
ssrd (𝜑𝐴𝐵)

Proof of Theorem ssrd
StepHypRef Expression
1 ssrd.0 . . 3 𝑥𝜑
2 ssrd.3 . . 3 (𝜑 → (𝑥𝐴𝑥𝐵))
31, 2alrimi 2212 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))
4 ssrd.1 . . 3 𝑥𝐴
5 ssrd.2 . . 3 𝑥𝐵
64, 5dfss2f 3908 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
73, 6sylibr 237 1 (𝜑𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  Ⅎwnf 1785   ∈ wcel 2112  Ⅎwnfc 2939   ⊆ wss 3884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-in 3891  df-ss 3901 This theorem is referenced by:  neiptopnei  21741  rabss3d  30289  topdifinffinlem  34765  relowlssretop  34781  ralssiun  34825  ssdf2  41771  ssfiunibd  41934  stoweidlem52  42687  stoweidlem59  42694
 Copyright terms: Public domain W3C validator