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Theorem elabg 2986
 Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
Hypothesis
Ref Expression
elabg.1 (x = A → (φψ))
Assertion
Ref Expression
elabg (A V → (A {x φ} ↔ ψ))
Distinct variable groups:   ψ,x   x,A
Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem elabg
StepHypRef Expression
1 nfcv 2489 . 2 xA
2 nfv 1619 . 2 xψ
3 elabg.1 . 2 (x = A → (φψ))
41, 2, 3elabgf 2983 1 (A V → (A {x φ} ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = 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-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:  elab2g  2987  intmin3  3954  peano5  4409  findsd  4410  nnadjoin  4520  spfininduct  4540  vfinspclt  4552  elxpi  4800  elimasn  5019  clos1induct  5880  clos1is  5881  elmapg  6012  elce  6175  spacis  6288
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