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Theorem rabssd 43831
Description: Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
rabssd.1 𝑥𝜑
rabssd.2 𝑥𝐵
rabssd.3 ((𝜑𝑥𝐴𝜒) → 𝑥𝐵)
Assertion
Ref Expression
rabssd (𝜑 → {𝑥𝐴𝜒} ⊆ 𝐵)

Proof of Theorem rabssd
StepHypRef Expression
1 rabssd.1 . . 3 𝑥𝜑
2 rabssd.3 . . . 4 ((𝜑𝑥𝐴𝜒) → 𝑥𝐵)
323exp 1120 . . 3 (𝜑 → (𝑥𝐴 → (𝜒𝑥𝐵)))
41, 3ralrimi 3255 . 2 (𝜑 → ∀𝑥𝐴 (𝜒𝑥𝐵))
5 rabssd.2 . . 3 𝑥𝐵
65rabssf 43808 . 2 ({𝑥𝐴𝜒} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜒𝑥𝐵))
74, 6sylibr 233 1 (𝜑 → {𝑥𝐴𝜒} ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088  wnf 1786  wcel 2107  wnfc 2884  wral 3062  {crab 3433  wss 3949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966
This theorem is referenced by:  pimxrneun  44199  fsupdm  45558  finfdm  45562
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