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Theorem rabssd 42729
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 1119 . . 3 (𝜑 → (𝑥𝐴 → (𝜒𝑥𝐵)))
41, 3ralrimi 3237 . 2 (𝜑 → ∀𝑥𝐴 (𝜒𝑥𝐵))
5 rabssd.2 . . 3 𝑥𝐵
65rabssf 42706 . 2 ({𝑥𝐴𝜒} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜒𝑥𝐵))
74, 6sylibr 233 1 (𝜑 → {𝑥𝐴𝜒} ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wnf 1783  wcel 2104  wnfc 2885  wral 3062  {crab 3284  wss 3892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rab 3287  df-v 3439  df-in 3899  df-ss 3909
This theorem is referenced by:  pimxrneun  43077
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