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Theorem elab3 2992
 Description: Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
Hypotheses
Ref Expression
elab3.1 (ψA V)
elab3.2 (x = A → (φψ))
Assertion
Ref Expression
elab3 (A {x φ} ↔ ψ)
Distinct variable groups:   ψ,x   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem elab3
StepHypRef Expression
1 elab3.1 . 2 (ψA V)
2 elab3.2 . . 3 (x = A → (φψ))
32elab3g 2991 . 2 ((ψA V) → (A {x φ} ↔ ψ))
41, 3ax-mp 8 1 (A {x φ} ↔ ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-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 This theorem is referenced by:  fvelrnb  5365  ovelrn  5608
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