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Theorem elab2 2988
 Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
Hypotheses
Ref Expression
elab2.1 A V
elab2.2 (x = A → (φψ))
elab2.3 B = {x φ}
Assertion
Ref Expression
elab2 (A Bψ)
Distinct variable groups:   ψ,x   x,A
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem elab2
StepHypRef Expression
1 elab2.1 . 2 A V
2 elab2.2 . . 3 (x = A → (φψ))
3 elab2.3 . . 3 B = {x φ}
42, 3elab2g 2987 . 2 (A V → (A Bψ))
51, 4ax-mp 8 1 (A Bψ)
 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:  elpw  3728  elint  3932  opkelopkabg  4245  0ceven  4505  eventfin  4517  oddtfin  4518  dfphi2  4569  phi11lem1  4595  0cnelphi  4597  proj1op  4600  proj2op  4601  opabid  4695  oprabid  5550  elfuns  5829
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