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Theorem ssdf 38729
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 450 . . 3 (𝜑 → (𝑥𝐴𝑥𝐵))
41, 3ralrimi 2951 . 2 (𝜑 → ∀𝑥𝐴 𝑥𝐵)
5 dfss3 3573 . 2 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
64, 5sylibr 224 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wnf 1705  wcel 1987  wral 2907  wss 3555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-in 3562  df-ss 3569
This theorem is referenced by:  ssd  38734  smfaddlem2  40276  smfadd  40277  smfmullem4  40305  smfmul  40306  smflimsuplem4  40333
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