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Theorem i3th5 547
 Description: Theorem for Kalmbach implication. (Contributed by NM, 16-Nov-1997.)
Assertion
Ref Expression
i3th5 ((a3 b) →3 (a3 (a3 b))) = 1

Proof of Theorem i3th5
StepHypRef Expression
1 ax-a2 31 . . . . . 6 ((ab) ∪ (ab )) = ((ab ) ∪ (ab))
2 lea 160 . . . . . . 7 (ab ) ≤ a
3 lear 161 . . . . . . 7 (ab) ≤ b
42, 3le2or 168 . . . . . 6 ((ab ) ∪ (ab)) ≤ (ab)
51, 4bltr 138 . . . . 5 ((ab) ∪ (ab )) ≤ (ab)
6 lear 161 . . . . 5 (a ∩ (ab)) ≤ (ab)
75, 6le2or 168 . . . 4 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) ≤ ((ab) ∪ (ab))
8 oridm 110 . . . 4 ((ab) ∪ (ab)) = (ab)
97, 8lbtr 139 . . 3 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) ≤ (ab)
10 df-i3 46 . . 3 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
11 lem4 511 . . 3 (a3 (a3 b)) = (ab)
129, 10, 11le3tr1 140 . 2 (a3 b) ≤ (a3 (a3 b))
1312lei3 246 1 ((a3 b) →3 (a3 (a3 b))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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