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Theorem dfin4 3495
 Description: Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.)
Assertion
Ref Expression
dfin4 (AB) = (A (A B))

Proof of Theorem dfin4
StepHypRef Expression
1 inss1 3475 . . 3 (AB) A
2 dfss4 3489 . . 3 ((AB) A ↔ (A (A (AB))) = (AB))
31, 2mpbi 199 . 2 (A (A (AB))) = (AB)
4 difin 3492 . . 3 (A (AB)) = (A B)
54difeq2i 3382 . 2 (A (A (AB))) = (A (A B))
63, 5eqtr3i 2375 1 (AB) = (A (A B))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∖ cdif 3206   ∩ cin 3208   ⊆ wss 3257 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-dif 3215  df-ss 3259 This theorem is referenced by:  indif  3497  cnvin  5035  imain  5172  resin  5307
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