New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  rabss2 GIF version

Theorem rabss2 3349
 Description: Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
rabss2 (A B → {x A φ} {x B φ})
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem rabss2
StepHypRef Expression
1 pm3.45 807 . . . 4 ((x Ax B) → ((x A φ) → (x B φ)))
21alimi 1559 . . 3 (x(x Ax B) → x((x A φ) → (x B φ)))
3 dfss2 3262 . . 3 (A Bx(x Ax B))
4 ss2ab 3334 . . 3 ({x (x A φ)} {x (x B φ)} ↔ x((x A φ) → (x B φ)))
52, 3, 43imtr4i 257 . 2 (A B → {x (x A φ)} {x (x B φ)})
6 df-rab 2623 . 2 {x A φ} = {x (x A φ)}
7 df-rab 2623 . 2 {x B φ} = {x (x B φ)}
85, 6, 73sstr4g 3312 1 (A B → {x A φ} {x B φ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  {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-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator