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Theorem imp3 841
 Description: Implicational product with 3 variables. Theorem 3.20 of "Equations, states, and lattices..." paper.
Assertion
Ref Expression
imp3 ((a2 b) ∩ (b1 c)) = ((ab ) ∪ (bc))

Proof of Theorem imp3
StepHypRef Expression
1 df-i1 44 . . 3 (b1 c) = (b ∪ (bc))
21lan 77 . 2 ((a2 b) ∩ (b1 c)) = ((a2 b) ∩ (b ∪ (bc)))
3 u2lemc1 681 . . . 4 b C (a2 b)
43comcom3 454 . . 3 b C (a2 b)
5 comanr1 464 . . . 4 b C (bc)
65comcom3 454 . . 3 b C (bc)
74, 6fh2 470 . 2 ((a2 b) ∩ (b ∪ (bc))) = (((a2 b) ∩ b ) ∪ ((a2 b) ∩ (bc)))
8 u2lemanb 616 . . 3 ((a2 b) ∩ b ) = (ab )
9 ancom 74 . . . 4 ((a2 b) ∩ (bc)) = ((bc) ∩ (a2 b))
10 lea 160 . . . . . 6 (bc) ≤ b
11 u2lem3 750 . . . . . . 7 (b2 (a2 b)) = 1
1211u2lemle2 716 . . . . . 6 b ≤ (a2 b)
1310, 12letr 137 . . . . 5 (bc) ≤ (a2 b)
1413df2le2 136 . . . 4 ((bc) ∩ (a2 b)) = (bc)
159, 14ax-r2 36 . . 3 ((a2 b) ∩ (bc)) = (bc)
168, 152or 72 . 2 (((a2 b) ∩ b ) ∪ ((a2 b) ∩ (bc))) = ((ab ) ∪ (bc))
172, 7, 163tr 65 1 ((a2 b) ∩ (b1 c)) = ((ab ) ∪ (bc))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   →2 wi2 13 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:  orbi  842  mlaconj4  844  mhcor1  888
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