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Theorem elrabf 2673
 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 Bφ} ↔ (A B ψ))

Proof of Theorem elrabf
StepHypRef Expression
1 elex 2543 . 2 (A {x Bφ} → A V)
2 elex 2543 . . 3 (A BA V)
32adantr 261 . 2 ((A B ψ) → A V)
4 df-rab 2293 . . . 4 {x Bφ} = {x ∣ (x B φ)}
54eleq2i 2086 . . 3 (A {x Bφ} ↔ A {x ∣ (x B φ)})
6 elrabf.1 . . . 4 xA
7 elrabf.2 . . . . . 6 xB
86, 7nfel 2168 . . . . 5 x A B
9 elrabf.3 . . . . 5 xψ
108, 9nfan 1439 . . . 4 x(A B ψ)
11 eleq1 2082 . . . . 5 (x = A → (x BA B))
12 elrabf.4 . . . . 5 (x = A → (φψ))
1311, 12anbi12d 445 . . . 4 (x = A → ((x B φ) ↔ (A B ψ)))
146, 10, 13elabgf 2662 . . 3 (A V → (A {x ∣ (x B φ)} ↔ (A B ψ)))
155, 14syl5bb 181 . 2 (A V → (A {x Bφ} ↔ (A B ψ)))
161, 3, 15pm5.21nii 607 1 (A {x Bφ} ↔ (A B ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228  Ⅎwnf 1329   ∈ wcel 1374  {cab 2008  Ⅎwnfc 2147  {crab 2288  Vcvv 2535 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rab 2293  df-v 2537 This theorem is referenced by:  elrab  2675  rabxfrd  4151
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