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Theorem ssdf 39738
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 3087 . 2 (𝜑 → ∀𝑥𝐴 𝑥𝐵)
5 dfss3 3725 . 2 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
64, 5sylibr 224 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wnf 1849  wcel 2131  wral 3042  wss 3707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-ral 3047  df-in 3714  df-ss 3721
This theorem is referenced by:  ssd  39743  smfaddlem2  41470  smfadd  41471  smfmullem4  41499  smfmul  41500  smflimsuplem4  41527
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