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Theorem ssrab 3344
 Description: Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
ssrab (B {x A φ} ↔ (B A x B φ))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem ssrab
StepHypRef Expression
1 df-rab 2623 . . 3 {x A φ} = {x (x A φ)}
21sseq2i 3296 . 2 (B {x A φ} ↔ B {x (x A φ)})
3 ssab 3336 . 2 (B {x (x A φ)} ↔ x(x B → (x A φ)))
4 dfss3 3263 . . . 4 (B Ax B x A)
54anbi1i 676 . . 3 ((B A x B φ) ↔ (x B x A x B φ))
6 r19.26 2746 . . 3 (x B (x A φ) ↔ (x B x A x B φ))
7 df-ral 2619 . . 3 (x B (x A φ) ↔ x(x B → (x A φ)))
85, 6, 73bitr2ri 265 . 2 (x(x B → (x A φ)) ↔ (B A x B φ))
92, 3, 83bitri 262 1 (B {x A φ} ↔ (B A x B φ))
 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-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-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:  ssrabdv  3345
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