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Theorem distid 887
 Description: Distributive law for identity.
Assertion
Ref Expression
distid ((ab) ∩ ((ac) ∪ (bc))) = (((ab) ∩ (ac)) ∪ ((ab) ∩ (bc)))

Proof of Theorem distid
StepHypRef Expression
1 lea 160 . . . 4 ((ab) ∩ ((ac) ∪ (bc))) ≤ (ab)
2 mlaconjo 886 . . . 4 ((ab) ∩ ((ac) ∪ (bc))) ≤ (ac)
31, 2ler2an 173 . . 3 ((ab) ∩ ((ac) ∪ (bc))) ≤ ((ab) ∩ (ac))
4 bicom 96 . . . . . 6 (ab) = (ba)
5 ax-a2 31 . . . . . 6 ((ac) ∪ (bc)) = ((bc) ∪ (ac))
64, 52an 79 . . . . 5 ((ab) ∩ ((ac) ∪ (bc))) = ((ba) ∩ ((bc) ∪ (ac)))
7 mlaconjo 886 . . . . 5 ((ba) ∩ ((bc) ∪ (ac))) ≤ (bc)
86, 7bltr 138 . . . 4 ((ab) ∩ ((ac) ∪ (bc))) ≤ (bc)
91, 8ler2an 173 . . 3 ((ab) ∩ ((ac) ∪ (bc))) ≤ ((ab) ∩ (bc))
103, 9ler2or 172 . 2 ((ab) ∩ ((ac) ∪ (bc))) ≤ (((ab) ∩ (ac)) ∪ ((ab) ∩ (bc)))
11 ledi 174 . 2 (((ab) ∩ (ac)) ∪ ((ab) ∩ (bc))) ≤ ((ab) ∩ ((ac) ∪ (bc)))
1210, 11lebi 145 1 ((ab) ∩ ((ac) ∪ (bc))) = (((ab) ∩ (ac)) ∪ ((ab) ∩ (bc)))
 Colors of variables: term Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6   ∩ wa 7 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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