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Theorem ssdf2 44507
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 3985 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1778  wcel 2099  wnfc 2879  wss 3947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-v 3473  df-in 3954  df-ss 3964
This theorem is referenced by:  supminfxr2  44851  fsupdm  46230  finfdm  46234
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