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Theorem ssdf 44078
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 412 . . 3 (𝜑 → (𝑥𝐴𝑥𝐵))
41, 3ralrimi 3253 . 2 (𝜑 → ∀𝑥𝐴 𝑥𝐵)
5 dfss3 3970 . 2 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
64, 5sylibr 233 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1784  wcel 2105  wral 3060  wss 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-v 3475  df-in 3955  df-ss 3965
This theorem is referenced by:  ssd  44083  smfaddlem2  45791  smfadd  45792  smfmullem4  45821  smfmul  45822  smflimsuplem4  45850
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