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Theorem difin2 3176
 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 2916 . . . . 5 (A𝐶 → (x Ax 𝐶))
21pm4.71d 373 . . . 4 (A𝐶 → (x A ↔ (x A x 𝐶)))
32anbi1d 441 . . 3 (A𝐶 → ((x A ¬ x B) ↔ ((x A x 𝐶) ¬ x B)))
4 eldif 2904 . . 3 (x (AB) ↔ (x A ¬ x B))
5 elin 3103 . . . 4 (x ((𝐶B) ∩ A) ↔ (x (𝐶B) x A))
6 eldif 2904 . . . . 5 (x (𝐶B) ↔ (x 𝐶 ¬ x B))
76anbi1i 434 . . . 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 383 . . . . 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 2020 1 (A𝐶 → (AB) = ((𝐶B) ∩ A))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374   ∖ cdif 2891   ∩ cin 2893   ⊆ wss 2894 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-in 2901  df-ss 2908 This theorem is referenced by: (None)
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