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Theorem indifdir 3187
Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
Assertion
Ref Expression
indifdir ((AB) ∩ 𝐶) = ((A𝐶) ∖ (B𝐶))

Proof of Theorem indifdir
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elin 3120 . . . 4 (x (A𝐶) ↔ (x A x 𝐶))
2 elin 3120 . . . . 5 (x (B𝐶) ↔ (x B x 𝐶))
32notbii 593 . . . 4 x (B𝐶) ↔ ¬ (x B x 𝐶))
41, 3anbi12i 433 . . 3 ((x (A𝐶) ¬ x (B𝐶)) ↔ ((x A x 𝐶) ¬ (x B x 𝐶)))
5 eldif 2921 . . 3 (x ((A𝐶) ∖ (B𝐶)) ↔ (x (A𝐶) ¬ x (B𝐶)))
6 elin 3120 . . . . 5 (x ((AB) ∩ 𝐶) ↔ (x (AB) x 𝐶))
7 eldif 2921 . . . . . 6 (x (AB) ↔ (x A ¬ x B))
87anbi1i 431 . . . . 5 ((x (AB) x 𝐶) ↔ ((x A ¬ x B) x 𝐶))
96, 8bitri 173 . . . 4 (x ((AB) ∩ 𝐶) ↔ ((x A ¬ x B) x 𝐶))
10 an32 496 . . . . 5 (((x A ¬ x B) x 𝐶) ↔ ((x A x 𝐶) ¬ x B))
11 simpl 102 . . . . . . . 8 ((x B x 𝐶) → x B)
1211con3i 561 . . . . . . 7 x B → ¬ (x B x 𝐶))
1312anim2i 324 . . . . . 6 (((x A x 𝐶) ¬ x B) → ((x A x 𝐶) ¬ (x B x 𝐶)))
14 simpl 102 . . . . . . 7 (((x A x 𝐶) ¬ (x B x 𝐶)) → (x A x 𝐶))
15 ax-in2 545 . . . . . . . . . . 11 (¬ (x B x 𝐶) → ((x B x 𝐶) → ⊥ ))
1615expcomd 1327 . . . . . . . . . 10 (¬ (x B x 𝐶) → (x 𝐶 → (x B → ⊥ )))
1716impcom 116 . . . . . . . . 9 ((x 𝐶 ¬ (x B x 𝐶)) → (x B → ⊥ ))
18 dfnot 1261 . . . . . . . . 9 x B ↔ (x B → ⊥ ))
1917, 18sylibr 137 . . . . . . . 8 ((x 𝐶 ¬ (x B x 𝐶)) → ¬ x B)
2019adantll 445 . . . . . . 7 (((x A x 𝐶) ¬ (x B x 𝐶)) → ¬ x B)
2114, 20jca 290 . . . . . 6 (((x A x 𝐶) ¬ (x B x 𝐶)) → ((x A x 𝐶) ¬ x B))
2213, 21impbii 117 . . . . 5 (((x A x 𝐶) ¬ x B) ↔ ((x A x 𝐶) ¬ (x B x 𝐶)))
2310, 22bitri 173 . . . 4 (((x A ¬ x B) x 𝐶) ↔ ((x A x 𝐶) ¬ (x B x 𝐶)))
249, 23bitri 173 . . 3 (x ((AB) ∩ 𝐶) ↔ ((x A x 𝐶) ¬ (x B x 𝐶)))
254, 5, 243bitr4ri 202 . 2 (x ((AB) ∩ 𝐶) ↔ x ((A𝐶) ∖ (B𝐶)))
2625eqriv 2034 1 ((AB) ∩ 𝐶) = ((A𝐶) ∖ (B𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242  wfal 1247   wcel 1390  cdif 2908  cin 2910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918
This theorem is referenced by: (None)
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