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Theorem difin0ss 3616
 Description: Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
Assertion
Ref Expression
difin0ss (((A B) ∩ C) = → (C AC B))

Proof of Theorem difin0ss
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eq0 3564 . 2 (((A B) ∩ C) = x ¬ x ((A B) ∩ C))
2 iman 413 . . . . . 6 ((x C → (x Ax B)) ↔ ¬ (x C ¬ (x Ax B)))
3 elin 3219 . . . . . . . 8 (x ((A B) ∩ C) ↔ (x (A B) x C))
4 eldif 3221 . . . . . . . . 9 (x (A B) ↔ (x A ¬ x B))
54anbi1i 676 . . . . . . . 8 ((x (A B) x C) ↔ ((x A ¬ x B) x C))
63, 5bitri 240 . . . . . . 7 (x ((A B) ∩ C) ↔ ((x A ¬ x B) x C))
7 ancom 437 . . . . . . 7 ((x C (x A ¬ x B)) ↔ ((x A ¬ x B) x C))
8 annim 414 . . . . . . . 8 ((x A ¬ x B) ↔ ¬ (x Ax B))
98anbi2i 675 . . . . . . 7 ((x C (x A ¬ x B)) ↔ (x C ¬ (x Ax B)))
106, 7, 93bitr2i 264 . . . . . 6 (x ((A B) ∩ C) ↔ (x C ¬ (x Ax B)))
112, 10xchbinxr 302 . . . . 5 ((x C → (x Ax B)) ↔ ¬ x ((A B) ∩ C))
12 ax-2 6 . . . . 5 ((x C → (x Ax B)) → ((x Cx A) → (x Cx B)))
1311, 12sylbir 204 . . . 4 x ((A B) ∩ C) → ((x Cx A) → (x Cx B)))
1413al2imi 1561 . . 3 (x ¬ x ((A B) ∩ C) → (x(x Cx A) → x(x Cx B)))
15 dfss2 3262 . . 3 (C Ax(x Cx A))
16 dfss2 3262 . . 3 (C Bx(x Cx B))
1714, 15, 163imtr4g 261 . 2 (x ¬ x ((A B) ∩ C) → (C AC B))
181, 17sylbi 187 1 (((A B) ∩ C) = → (C AC B))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550 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-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by: (None)
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