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Theorem ssundif 3633
 Description: A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.)
Assertion
Ref Expression
ssundif (A (BC) ↔ (A B) C)

Proof of Theorem ssundif
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 pm5.6 878 . . . 4 (((x A ¬ x B) → x C) ↔ (x A → (x B x C)))
2 eldif 3221 . . . . 5 (x (A B) ↔ (x A ¬ x B))
32imbi1i 315 . . . 4 ((x (A B) → x C) ↔ ((x A ¬ x B) → x C))
4 elun 3220 . . . . 5 (x (BC) ↔ (x B x C))
54imbi2i 303 . . . 4 ((x Ax (BC)) ↔ (x A → (x B x C)))
61, 3, 53bitr4ri 269 . . 3 ((x Ax (BC)) ↔ (x (A B) → x C))
76albii 1566 . 2 (x(x Ax (BC)) ↔ x(x (A B) → x C))
8 dfss2 3262 . 2 (A (BC) ↔ x(x Ax (BC)))
9 dfss2 3262 . 2 ((A B) Cx(x (A B) → x C))
107, 8, 93bitr4i 268 1 (A (BC) ↔ (A B) C)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540   ∈ wcel 1710   ∖ cdif 3206   ∪ cun 3207   ⊆ wss 3257 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259 This theorem is referenced by:  difcom  3634  uneqdifeq  3638  ssunsn2  3865  pwadjoin  4119
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