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Theorem ssrd 3987
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, 5dfssf 3973 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
73, 6sylibr 234 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1537  wnf 1782  wcel 2107  wnfc 2889  wss 3950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-11 2156  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-clel 2815  df-nfc 2891  df-ss 3967
This theorem is referenced by:  rabss3d  4080  neiptopnei  23141  topdifinffinlem  37349  relowlssretop  37365  ralssiun  37409  sticksstones1  42148  sticksstones11  42158  ssdf2  45151  ssfiunibd  45326  stoweidlem52  46072  stoweidlem59  46079
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