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Theorem indifcom 3500
 Description: Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
Assertion
Ref Expression
indifcom (A ∩ (B C)) = (B ∩ (A C))

Proof of Theorem indifcom
StepHypRef Expression
1 incom 3448 . . 3 (AB) = (BA)
21difeq1i 3381 . 2 ((AB) C) = ((BA) C)
3 indif2 3498 . 2 (A ∩ (B C)) = ((AB) C)
4 indif2 3498 . 2 (B ∩ (A C)) = ((BA) C)
52, 3, 43eqtr4i 2383 1 (A ∩ (B C)) = (B ∩ (A C))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∖ cdif 3206   ∩ 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-dif 3215 This theorem is referenced by: (None)
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