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Theorem ssdf 42186
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 416 . . 3 (𝜑 → (𝑥𝐴𝑥𝐵))
41, 3ralrimi 3129 . 2 (𝜑 → ∀𝑥𝐴 𝑥𝐵)
5 dfss3 3866 . 2 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
64, 5sylibr 237 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wnf 1790  wcel 2114  wral 3054  wss 3844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-12 2179  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-v 3401  df-in 3851  df-ss 3861
This theorem is referenced by:  ssd  42191  smfaddlem2  43861  smfadd  43862  smfmullem4  43890  smfmul  43891  smflimsuplem4  43918
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