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Theorem ssrd 3980
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 2198 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))
4 ssrd.1 . . 3 𝑥𝐴
5 ssrd.2 . . 3 𝑥𝐵
64, 5dfss2f 3965 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
73, 6sylibr 233 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  wnf 1777  wcel 2098  wnfc 2875  wss 3941
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-v 3468  df-in 3948  df-ss 3958
This theorem is referenced by:  rabss3d  4072  neiptopnei  22960  topdifinffinlem  36719  relowlssretop  36735  ralssiun  36779  sticksstones1  41459  sticksstones11  41469  ssdf2  44343  ssfiunibd  44529  stoweidlem52  45278  stoweidlem59  45285
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