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Theorem difin2 3193
Description: Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
difin2 (A𝐶 → (AB) = ((𝐶B) ∩ A))

Proof of Theorem difin2
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ssel 2933 . . . . 5 (A𝐶 → (x Ax 𝐶))
21pm4.71d 373 . . . 4 (A𝐶 → (x A ↔ (x A x 𝐶)))
32anbi1d 438 . . 3 (A𝐶 → ((x A ¬ x B) ↔ ((x A x 𝐶) ¬ x B)))
4 eldif 2921 . . 3 (x (AB) ↔ (x A ¬ x B))
5 elin 3120 . . . 4 (x ((𝐶B) ∩ A) ↔ (x (𝐶B) x A))
6 eldif 2921 . . . . 5 (x (𝐶B) ↔ (x 𝐶 ¬ x B))
76anbi1i 431 . . . 4 ((x (𝐶B) x A) ↔ ((x 𝐶 ¬ x B) x A))
8 ancom 253 . . . . 5 (((x 𝐶 ¬ x B) x A) ↔ (x A (x 𝐶 ¬ x B)))
9 anass 381 . . . . 5 (((x A x 𝐶) ¬ x B) ↔ (x A (x 𝐶 ¬ x B)))
108, 9bitr4i 176 . . . 4 (((x 𝐶 ¬ x B) x A) ↔ ((x A x 𝐶) ¬ x B))
115, 7, 103bitri 195 . . 3 (x ((𝐶B) ∩ A) ↔ ((x A x 𝐶) ¬ x B))
123, 4, 113bitr4g 212 . 2 (A𝐶 → (x (AB) ↔ x ((𝐶B) ∩ A)))
1312eqrdv 2035 1 (A𝐶 → (AB) = ((𝐶B) ∩ A))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242   wcel 1390  cdif 2908  cin 2910  wss 2911
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-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  df-ss 2925
This theorem is referenced by: (None)
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