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Theorem in4 3471
 Description: Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
Assertion
Ref Expression
in4 ((AB) ∩ (CD)) = ((AC) ∩ (BD))

Proof of Theorem in4
StepHypRef Expression
1 in12 3466 . . 3 (B ∩ (CD)) = (C ∩ (BD))
21ineq2i 3454 . 2 (A ∩ (B ∩ (CD))) = (A ∩ (C ∩ (BD)))
3 inass 3465 . 2 ((AB) ∩ (CD)) = (A ∩ (B ∩ (CD)))
4 inass 3465 . 2 ((AC) ∩ (BD)) = (A ∩ (C ∩ (BD)))
52, 3, 43eqtr4i 2383 1 ((AB) ∩ (CD)) = ((AC) ∩ (BD))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∩ 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 This theorem is referenced by:  inindi  3472  inindir  3473
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