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Theorem undif3 3515
 Description: An equality involving class union and class difference. The first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 17-Apr-2012.)
Assertion
Ref Expression
undif3 (A ∪ (B C)) = ((AB) (C A))

Proof of Theorem undif3
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elun 3220 . . . 4 (x (AB) ↔ (x A x B))
2 pm4.53 478 . . . . 5 (¬ (x C ¬ x A) ↔ (¬ x C x A))
3 eldif 3221 . . . . 5 (x (C A) ↔ (x C ¬ x A))
42, 3xchnxbir 300 . . . 4 x (C A) ↔ (¬ x C x A))
51, 4anbi12i 678 . . 3 ((x (AB) ¬ x (C A)) ↔ ((x A x B) x C x A)))
6 eldif 3221 . . 3 (x ((AB) (C A)) ↔ (x (AB) ¬ x (C A)))
7 elun 3220 . . . 4 (x (A ∪ (B C)) ↔ (x A x (B C)))
8 eldif 3221 . . . . 5 (x (B C) ↔ (x B ¬ x C))
98orbi2i 505 . . . 4 ((x A x (B C)) ↔ (x A (x B ¬ x C)))
10 orc 374 . . . . . . 7 (x A → (x A x B))
11 olc 373 . . . . . . 7 (x A → (¬ x C x A))
1210, 11jca 518 . . . . . 6 (x A → ((x A x B) x C x A)))
13 olc 373 . . . . . . 7 (x B → (x A x B))
14 orc 374 . . . . . . 7 x C → (¬ x C x A))
1513, 14anim12i 549 . . . . . 6 ((x B ¬ x C) → ((x A x B) x C x A)))
1612, 15jaoi 368 . . . . 5 ((x A (x B ¬ x C)) → ((x A x B) x C x A)))
17 simpl 443 . . . . . . 7 ((x A ¬ x C) → x A)
1817orcd 381 . . . . . 6 ((x A ¬ x C) → (x A (x B ¬ x C)))
19 olc 373 . . . . . 6 ((x B ¬ x C) → (x A (x B ¬ x C)))
20 orc 374 . . . . . . 7 (x A → (x A (x B ¬ x C)))
2120adantr 451 . . . . . 6 ((x A x A) → (x A (x B ¬ x C)))
2220adantl 452 . . . . . 6 ((x B x A) → (x A (x B ¬ x C)))
2318, 19, 21, 22ccase 912 . . . . 5 (((x A x B) x C x A)) → (x A (x B ¬ x C)))
2416, 23impbii 180 . . . 4 ((x A (x B ¬ x C)) ↔ ((x A x B) x C x A)))
257, 9, 243bitri 262 . . 3 (x (A ∪ (B C)) ↔ ((x A x B) x C x A)))
265, 6, 253bitr4ri 269 . 2 (x (A ∪ (B C)) ↔ x ((AB) (C A)))
2726eqriv 2350 1 (A ∪ (B C)) = ((AB) (C A))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∪ cun 3207 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-un 3214  df-dif 3215 This theorem is referenced by:  undifabs  3627
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