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Theorem i3lem4 507
 Description: Lemma for Kalmbach implication.
Hypothesis
Ref Expression
i3lem.1 (a3 b) = 1
Assertion
Ref Expression
i3lem4 (ab) = 1

Proof of Theorem i3lem4
StepHypRef Expression
1 i3lem.1 . . . . 5 (a3 b) = 1
21i3lem1 504 . . . 4 ((ab) ∪ (ab )) = a
32ax-r5 38 . . 3 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) = (a ∪ (a ∩ (ab)))
43ax-r1 35 . 2 (a ∪ (a ∩ (ab))) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
5 omln 446 . 2 (a ∪ (a ∩ (ab))) = (ab)
6 df-i3 46 . . . 4 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
76ax-r1 35 . . 3 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) = (a3 b)
87, 1ax-r2 36 . 2 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) = 1
94, 5, 83tr2 64 1 (ab) = 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:  i3le  515
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