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Theorem ssdf2 45081
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 412 . 2 (𝜑 → (𝑥𝐴𝑥𝐵))
61, 2, 3, 5ssrd 4000 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1780  wcel 2106  wnfc 2888  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-clel 2814  df-nfc 2890  df-ss 3980
This theorem is referenced by:  supminfxr2  45419  fsupdm  46798  finfdm  46802
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