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Theorem elrabf 2690
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 2560 . 2 (A {x Bφ} → A V)
2 elex 2560 . . 3 (A BA V)
32adantr 261 . 2 ((A B ψ) → A V)
4 df-rab 2309 . . . 4 {x Bφ} = {x ∣ (x B φ)}
54eleq2i 2101 . . 3 (A {x Bφ} ↔ A {x ∣ (x B φ)})
6 elrabf.1 . . . 4 xA
7 elrabf.2 . . . . . 6 xB
86, 7nfel 2183 . . . . 5 x A B
9 elrabf.3 . . . . 5 xψ
108, 9nfan 1454 . . . 4 x(A B ψ)
11 eleq1 2097 . . . . 5 (x = A → (x BA B))
12 elrabf.4 . . . . 5 (x = A → (φψ))
1311, 12anbi12d 442 . . . 4 (x = A → ((x B φ) ↔ (A B ψ)))
146, 10, 13elabgf 2679 . . 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 619 1 (A {x Bφ} ↔ (A B ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wnf 1346   wcel 1390  {cab 2023  wnfc 2162  {crab 2304  Vcvv 2551
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553
This theorem is referenced by:  elrab  2692  rabxfrd  4166
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