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Theorem comanb 872
 Description: Biconditional commutation law.
Assertion
Ref Expression
comanb (ab) C ((ac) ∩ (bc))

Proof of Theorem comanb
StepHypRef Expression
1 lea 160 . . . 4 (((ac) ∪ ((ab) ∩ c)) ∩ (b1 c)) ≤ ((ac) ∪ ((ab) ∩ c))
2 lea 160 . . . . . . 7 (ab) ≤ a
3 leo 158 . . . . . . 7 a ≤ (ac)
42, 3letr 137 . . . . . 6 (ab) ≤ (ac)
54lecon 154 . . . . 5 (ac) ≤ (ab)
65leror 152 . . . 4 ((ac) ∪ ((ab) ∩ c)) ≤ ((ab) ∪ ((ab) ∩ c))
71, 6letr 137 . . 3 (((ac) ∪ ((ab) ∩ c)) ∩ (b1 c)) ≤ ((ab) ∪ ((ab) ∩ c))
8 comanblem1 870 . . 3 ((ac) ∩ (bc)) = (((ac) ∪ ((ab) ∩ c)) ∩ (b1 c))
9 df-i1 44 . . . 4 ((ab) →1 ((ac) ∩ (bc))) = ((ab) ∪ ((ab) ∩ ((ac) ∩ (bc))))
10 comanblem2 871 . . . . 5 ((ab) ∩ ((ac) ∩ (bc))) = ((ab) ∩ c)
1110lor 70 . . . 4 ((ab) ∪ ((ab) ∩ ((ac) ∩ (bc)))) = ((ab) ∪ ((ab) ∩ c))
129, 11ax-r2 36 . . 3 ((ab) →1 ((ac) ∩ (bc))) = ((ab) ∪ ((ab) ∩ c))
137, 8, 12le3tr1 140 . 2 ((ac) ∩ (bc)) ≤ ((ab) →1 ((ac) ∩ (bc)))
1413i1com 708 1 (ab) C ((ac) ∩ (bc))
 Colors of variables: term Syntax hints:   C wc 3  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →1 wi1 12 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-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  comanbn  873
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