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Theorem rabss 3343
 Description: Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
rabss ({x A φ} Bx A (φx B))
Distinct variable group:   x,B
Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem rabss
StepHypRef Expression
1 df-rab 2623 . . 3 {x A φ} = {x (x A φ)}
21sseq1i 3295 . 2 ({x A φ} B ↔ {x (x A φ)} B)
3 abss 3335 . 2 ({x (x A φ)} Bx((x A φ) → x B))
4 impexp 433 . . . 4 (((x A φ) → x B) ↔ (x A → (φx B)))
54albii 1566 . . 3 (x((x A φ) → x B) ↔ x(x A → (φx B)))
6 df-ral 2619 . . 3 (x A (φx B) ↔ x(x A → (φx B)))
75, 6bitr4i 243 . 2 (x((x A φ) → x B) ↔ x A (φx B))
82, 3, 73bitri 262 1 ({x A φ} Bx A (φx B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  ∀wral 2614  {crab 2618   ⊆ wss 3257 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-nan 1288  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-ral 2619  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  rabssdv  3346
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