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Theorem ssdf2 40087
Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
ssdf2.p 𝑥𝜑
ssdf2.a 𝑥𝐴
ssdf2.b 𝑥𝐵
ssdf2.x ((𝜑𝑥𝐴) → 𝑥𝐵)
Assertion
Ref Expression
ssdf2 (𝜑𝐴𝐵)

Proof of Theorem ssdf2
StepHypRef Expression
1 ssdf2.p . 2 𝑥𝜑
2 ssdf2.a . 2 𝑥𝐴
3 ssdf2.b . 2 𝑥𝐵
4 ssdf2.x . . 3 ((𝜑𝑥𝐴) → 𝑥𝐵)
54ex 402 . 2 (𝜑 → (𝑥𝐴𝑥𝐵))
61, 2, 3, 5ssrd 3803 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wnf 1879  wcel 2157  wnfc 2928  wss 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-in 3776  df-ss 3783
This theorem is referenced by:  supminfxr2  40442
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