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Theorem abbi2dv 2468
 Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
Hypothesis
Ref Expression
abbirdv.1 (φ → (x Aψ))
Assertion
Ref Expression
abbi2dv (φA = {x ψ})
Distinct variable groups:   x,A   φ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem abbi2dv
StepHypRef Expression
1 abbirdv.1 . . 3 (φ → (x Aψ))
21alrimiv 1631 . 2 (φx(x Aψ))
3 abeq2 2458 . 2 (A = {x ψ} ↔ x(x Aψ))
42, 3sylibr 203 1 (φA = {x ψ})
 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:  sbab  2475  iftrue  3668  iffalse  3669  phialllem1  4616  isoini  5497
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