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Theorem ssdf 38273
 Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
ssdf.1 𝑥𝜑
ssdf.2 ((𝜑𝑥𝐴) → 𝑥𝐵)
Assertion
Ref Expression
ssdf (𝜑𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssdf
StepHypRef Expression
1 ssdf.1 . . 3 𝑥𝜑
2 ssdf.2 . . . 4 ((𝜑𝑥𝐴) → 𝑥𝐵)
32ex 449 . . 3 (𝜑 → (𝑥𝐴𝑥𝐵))
41, 3ralrimi 2940 . 2 (𝜑 → ∀𝑥𝐴 𝑥𝐵)
5 dfss3 3558 . 2 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
64, 5sylibr 223 1 (𝜑𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896   ⊆ 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:  ssd  38278  smfaddlem2  39650  smfadd  39651  smfmullem4  39679  smfmul  39680
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