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Theorem elabf 2984
 Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
elabf.1 xψ
elabf.2 A V
elabf.3 (x = A → (φψ))
Assertion
Ref Expression
elabf (A {x φ} ↔ ψ)
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem elabf
StepHypRef Expression
1 elabf.2 . 2 A V
2 nfcv 2489 . . 3 xA
3 elabf.1 . . 3 xψ
4 elabf.3 . . 3 (x = A → (φψ))
52, 3, 4elabgf 2983 . 2 (A V → (A {x φ} ↔ ψ))
61, 5ax-mp 5 1 (A {x φ} ↔ ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859 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:  elab  2985
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