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Theorem elrabf 2993
 Description: Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)
Hypotheses
Ref Expression
elrabf.1 xA
elrabf.2 xB
elrabf.3 xψ
elrabf.4 (x = A → (φψ))
Assertion
Ref Expression
elrabf (A {x B φ} ↔ (A B ψ))

Proof of Theorem elrabf
StepHypRef Expression
1 elex 2867 . 2 (A {x B φ} → A V)
2 elex 2867 . . 3 (A BA V)
32adantr 451 . 2 ((A B ψ) → A V)
4 df-rab 2623 . . . 4 {x B φ} = {x (x B φ)}
54eleq2i 2417 . . 3 (A {x B φ} ↔ A {x (x B φ)})
6 elrabf.1 . . . 4 xA
7 elrabf.2 . . . . . 6 xB
86, 7nfel 2497 . . . . 5 x A B
9 elrabf.3 . . . . 5 xψ
108, 9nfan 1824 . . . 4 x(A B ψ)
11 eleq1 2413 . . . . 5 (x = A → (x BA B))
12 elrabf.4 . . . . 5 (x = A → (φψ))
1311, 12anbi12d 691 . . . 4 (x = A → ((x B φ) ↔ (A B ψ)))
146, 10, 13elabgf 2983 . . 3 (A V → (A {x (x B φ)} ↔ (A B ψ)))
155, 14syl5bb 248 . 2 (A V → (A {x B φ} ↔ (A B ψ)))
161, 3, 15pm5.21nii 342 1 (A {x B φ} ↔ (A B ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476  {crab 2618  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-rab 2623  df-v 2861 This theorem is referenced by:  elrab  2994
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