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Theorem ssdif 3401
 Description: Difference law for subsets. (Contributed by NM, 28-May-1998.)
Assertion
Ref Expression
ssdif (A B → (A C) (B C))

Proof of Theorem ssdif
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ssel 3267 . . . 4 (A B → (x Ax B))
21anim1d 547 . . 3 (A B → ((x A ¬ x C) → (x B ¬ x C)))
3 eldif 3221 . . 3 (x (A C) ↔ (x A ¬ x C))
4 eldif 3221 . . 3 (x (B C) ↔ (x B ¬ x C))
52, 3, 43imtr4g 261 . 2 (A B → (x (A C) → x (B C)))
65ssrdv 3278 1 (A B → (A C) (B C))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∈ wcel 1710   ∖ cdif 3206   ⊆ 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259 This theorem is referenced by:  ssdifd  3402  sbthlem1  6203
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