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Theorem abssdv 3638
Description: Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
Hypothesis
Ref Expression
abssdv.1 (𝜑 → (𝜓𝑥𝐴))
Assertion
Ref Expression
abssdv (𝜑 → {𝑥𝜓} ⊆ 𝐴)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem abssdv
StepHypRef Expression
1 abssdv.1 . . 3 (𝜑 → (𝜓𝑥𝐴))
21alrimiv 1841 . 2 (𝜑 → ∀𝑥(𝜓𝑥𝐴))
3 abss 3633 . 2 ({𝑥𝜓} ⊆ 𝐴 ↔ ∀𝑥(𝜓𝑥𝐴))
42, 3sylibr 222 1 (𝜑 → {𝑥𝜓} ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  wcel 1976  {cab 2595  wss 3539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-in 3546  df-ss 3553
This theorem is referenced by:  dfopif  4331  fmpt  6274  opabex2  6974  eroprf  7709  cfslb2n  8950  rankcf  9455  genpv  9677  genpdm  9680  fimaxre3  10819  supadd  10838  supmul  10842  hashfacen  13047  hashf1lem1  13048  hashf1lem2  13049  mertenslem2  14402  4sqlem11  15443  symgbas  17569  lss1d  18730  lspsn  18769  lpval  20695  lpsscls  20697  ptuni2  21131  ptbasfi  21136  prdstopn  21183  xkopt  21210  tgpconcompeqg  21667  metrest  22080  mbfeqalem  23132  limcfval  23359  nmosetre  26809  nmopsetretALT  27912  nmfnsetre  27926  sigaclcuni  29314  bnj849  30055  deranglem  30208  derangsn  30212  liness  31228  mblfinlem3  32414  ismblfin  32416  itg2addnclem  32427  areacirclem2  32467  sdclem2  32504  sdclem1  32505  ismtyval  32565  heibor1lem  32574  heibor1  32575  pmapglbx  33869  eldiophb  36134  hbtlem2  36509  upbdrech  38256
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