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Theorem elrabsf 3084
 Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2993 has implicit substitution). The hypothesis specifies that x must not be a free variable in B. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Hypothesis
Ref Expression
elrabsf.1 xB
Assertion
Ref Expression
elrabsf (A {x B φ} ↔ (A B A / xφ))

Proof of Theorem elrabsf
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3048 . 2 (y = A → ([̣y / xφ ↔ [̣A / xφ))
2 elrabsf.1 . . 3 xB
3 nfcv 2489 . . 3 yB
4 nfv 1619 . . 3 yφ
5 nfsbc1v 3065 . . 3 xy / xφ
6 sbceq1a 3056 . . 3 (x = y → (φ ↔ [̣y / xφ))
72, 3, 4, 5, 6cbvrab 2857 . 2 {x B φ} = {y B y / xφ}
81, 7elrab2 2996 1 (A {x B φ} ↔ (A B A / xφ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  Ⅎwnfc 2476  {crab 2618  [̣wsbc 3046 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-rab 2623  df-v 2861  df-sbc 3047 This theorem is referenced by: (None)
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