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Theorem dfin3 3494
 Description: Intersection defined in terms of union (De Morgan's law. Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
Assertion
Ref Expression
dfin3 (AB) = (V ((V A) ∪ (V B)))

Proof of Theorem dfin3
StepHypRef Expression
1 ddif 3398 . 2 (V (V (A (V B)))) = (A (V B))
2 dfun2 3490 . . . 4 ((V A) ∪ (V B)) = (V ((V (V A)) (V B)))
3 ddif 3398 . . . . . 6 (V (V A)) = A
43difeq1i 3381 . . . . 5 ((V (V A)) (V B)) = (A (V B))
54difeq2i 3382 . . . 4 (V ((V (V A)) (V B))) = (V (A (V B)))
62, 5eqtri 2373 . . 3 ((V A) ∪ (V B)) = (V (A (V B)))
76difeq2i 3382 . 2 (V ((V A) ∪ (V B))) = (V (V (A (V B))))
8 dfin2 3491 . 2 (AB) = (A (V B))
91, 7, 83eqtr4ri 2384 1 (AB) = (V ((V A) ∪ (V B)))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  Vcvv 2859   ∖ 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:  difindi  3509
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