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Theorem inabs 3486
 Description: Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
Assertion
Ref Expression
inabs (A ∩ (AB)) = A

Proof of Theorem inabs
StepHypRef Expression
1 ssun1 3426 . 2 A (AB)
2 df-ss 3259 . 2 (A (AB) ↔ (A ∩ (AB)) = A)
31, 2mpbi 199 1 (A ∩ (AB)) = A
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∪ cun 3207   ∩ cin 3208   ⊆ 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259 This theorem is referenced by:  dfif5  3674
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