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Theorem reldisj 3594
 Description: Two ways of saying that two classes are disjoint, using the complement of B relative to a universe C. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
reldisj (A C → ((AB) = A (C B)))

Proof of Theorem reldisj
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 dfss2 3262 . . . 4 (A Cx(x Ax C))
2 pm5.44 877 . . . . . 6 ((x Ax C) → ((x A → ¬ x B) ↔ (x A → (x C ¬ x B))))
3 eldif 3221 . . . . . . 7 (x (C B) ↔ (x C ¬ x B))
43imbi2i 303 . . . . . 6 ((x Ax (C B)) ↔ (x A → (x C ¬ x B)))
52, 4syl6bbr 254 . . . . 5 ((x Ax C) → ((x A → ¬ x B) ↔ (x Ax (C B))))
65sps 1754 . . . 4 (x(x Ax C) → ((x A → ¬ x B) ↔ (x Ax (C B))))
71, 6sylbi 187 . . 3 (A C → ((x A → ¬ x B) ↔ (x Ax (C B))))
87albidv 1625 . 2 (A C → (x(x A → ¬ x B) ↔ x(x Ax (C B))))
9 disj1 3593 . 2 ((AB) = x(x A → ¬ x B))
10 dfss2 3262 . 2 (A (C B) ↔ x(x Ax (C B)))
118, 9, 103bitr4g 279 1 (A C → ((AB) = A (C B)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by:  disj2  3598
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