New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  abbi1dv GIF version

Theorem abbi1dv 2469
 Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
Hypothesis
Ref Expression
abbildv.1 (φ → (ψx A))
Assertion
Ref Expression
abbi1dv (φ → {x ψ} = A)
Distinct variable groups:   x,A   φ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem abbi1dv
StepHypRef Expression
1 abbildv.1 . . 3 (φ → (ψx A))
21alrimiv 1631 . 2 (φx(ψx A))
3 abeq1 2459 . 2 ({x ψ} = Ax(ψx A))
42, 3sylibr 203 1 (φ → {x ψ} = A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349 This theorem is referenced by:  abidnf  3005  csbtt  3148  csbvarg  3163  csbie2g  3182  abvor0  3567  iinxsng  4042  enpw1  6062
 Copyright terms: Public domain W3C validator