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Theorem elrab2 2996
 Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
Hypotheses
Ref Expression
elrab2.1 (x = A → (φψ))
elrab2.2 C = {x B φ}
Assertion
Ref Expression
elrab2 (A C ↔ (A B ψ))
Distinct variable groups:   ψ,x   x,A   x,B
Allowed substitution hints:   φ(x)   C(x)

Proof of Theorem elrab2
StepHypRef Expression
1 elrab2.2 . . 3 C = {x B φ}
21eleq2i 2417 . 2 (A CA {x B φ})
3 elrab2.1 . . 3 (x = A → (φψ))
43elrab 2994 . 2 (A {x B φ} ↔ (A B ψ))
52, 4bitri 240 1 (A C ↔ (A B ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {crab 2618 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-rab 2623  df-v 2861 This theorem is referenced by:  elrabsf  3084  fvmpti  5699  fvmptss2  5725  nenpw1pwlem2  6085
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