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Theorem rabssdv 3667
Description: Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)
Hypothesis
Ref Expression
rabssdv.1 ((𝜑𝑥𝐴𝜓) → 𝑥𝐵)
Assertion
Ref Expression
rabssdv (𝜑 → {𝑥𝐴𝜓} ⊆ 𝐵)
Distinct variable groups:   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rabssdv
StepHypRef Expression
1 rabssdv.1 . . . 4 ((𝜑𝑥𝐴𝜓) → 𝑥𝐵)
213exp 1261 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝑥𝐵)))
32ralrimiv 2961 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝑥𝐵))
4 rabss 3664 . 2 ({𝑥𝐴𝜓} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜓𝑥𝐵))
53, 4sylibr 224 1 (𝜑 → {𝑥𝐴𝜓} ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036  wcel 1987  wral 2908  {crab 2912  wss 3560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rab 2917  df-in 3567  df-ss 3574
This theorem is referenced by:  suppss2  7289  oemapvali  8541  cantnflem1  8546  harval2  8783  zsupss  11737  ramub1lem1  15673  symggen  17830  efgsfo  18092  ablfacrp  18405  ablfac1eu  18412  pgpfac1lem5  18418  ablfaclem3  18426  nrmr0reg  21492  ptcmplem3  21798  abelthlem2  24124  lgamgulmlem1  24689  neibastop2lem  32050  topmeet  32054  cntotbnd  33266  mapdrvallem2  36453  k0004ss1  37970
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