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Theorem ssrd 3986
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 2206 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))
4 ssrd.1 . . 3 𝑥𝐴
5 ssrd.2 . . 3 𝑥𝐵
64, 5dfss2f 3971 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
73, 6sylibr 233 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1539  wnf 1785  wcel 2106  wnfc 2883  wss 3947
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-v 3476  df-in 3954  df-ss 3964
This theorem is referenced by:  rabss3d  4078  neiptopnei  22627  topdifinffinlem  36216  relowlssretop  36232  ralssiun  36276  sticksstones1  40950  sticksstones11  40960  ssdf2  43815  ssfiunibd  44005  stoweidlem52  44754  stoweidlem59  44761
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