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Theorem inundif 3628
 Description: The intersection and class difference of a class with another class unite to give the original class. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
inundif ((AB) ∪ (A B)) = A

Proof of Theorem inundif
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elin 3219 . . . 4 (x (AB) ↔ (x A x B))
2 eldif 3221 . . . 4 (x (A B) ↔ (x A ¬ x B))
31, 2orbi12i 507 . . 3 ((x (AB) x (A B)) ↔ ((x A x B) (x A ¬ x B)))
4 pm4.42 926 . . 3 (x A ↔ ((x A x B) (x A ¬ x B)))
53, 4bitr4i 243 . 2 ((x (AB) x (A B)) ↔ x A)
65uneqri 3406 1 ((AB) ∪ (A B)) = A
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208 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:  phialllem2  4617  sbthlem1  6203
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