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Theorem ssd 38278
 Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypothesis
Ref Expression
ssd.1 ((𝜑𝑥𝐴) → 𝑥𝐵)
Assertion
Ref Expression
ssd (𝜑𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Proof of Theorem ssd
StepHypRef Expression
1 nfv 1830 . 2 𝑥𝜑
2 ssd.1 . 2 ((𝜑𝑥𝐴) → 𝑥𝐵)
31, 2ssdf 38273 1 (𝜑𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-in 3547  df-ss 3554 This theorem is referenced by:  icomnfinre  38626  fnlimfvre  38741  allbutfifvre  38742  rrxsnicc  39196  meaiuninclem  39373  meaiininclem  39376  borelmbl  39526  smflimlem1  39657  smflimlem2  39658  smfpimbor1lem1  39683  smfpimbor1lem2  39684
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