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Theorem abssdv 3340
 Description: Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
Hypothesis
Ref Expression
abssdv.1 (φ → (ψx A))
Assertion
Ref Expression
abssdv (φ → {x ψ} A)
Distinct variable groups:   φ,x   x,A
Allowed substitution hint:   ψ(x)

Proof of Theorem abssdv
StepHypRef Expression
1 abssdv.1 . . 3 (φ → (ψx A))
21alrimiv 1631 . 2 (φx(ψx A))
3 abss 3335 . 2 ({x ψ} Ax(ψx A))
42, 3sylibr 203 1 (φ → {x ψ} A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 1710  {cab 2339   ⊆ wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  vfinspeqtncv  4553  fmpt  5692
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