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Theorem omlem1 127
 Description: Lemma in proof of Th. 1 of Pavicic 1987.
Assertion
Ref Expression
omlem1 ((a ∪ (a ∩ (ab))) ∪ (ab)) = (ab)

Proof of Theorem omlem1
StepHypRef Expression
1 ax-a2 31 . . 3 ((a ∪ (a ∩ (ab))) ∪ (ab)) = ((ab) ∪ (a ∪ (a ∩ (ab))))
2 ax-a3 32 . . 3 (((a ∪ (a ∩ (ab))) ∪ a) ∪ b) = ((a ∪ (a ∩ (ab))) ∪ (ab))
3 ax-a3 32 . . 3 (((ab) ∪ a) ∪ (a ∩ (ab))) = ((ab) ∪ (a ∪ (a ∩ (ab))))
41, 2, 33tr1 63 . 2 (((a ∪ (a ∩ (ab))) ∪ a) ∪ b) = (((ab) ∪ a) ∪ (a ∩ (ab)))
5 ax-a3 32 . . . . . . 7 ((aa) ∪ b) = (a ∪ (ab))
6 ax-a2 31 . . . . . . 7 (a ∪ (ab)) = ((ab) ∪ a)
75, 6ax-r2 36 . . . . . 6 ((aa) ∪ b) = ((ab) ∪ a)
87ax-r1 35 . . . . 5 ((ab) ∪ a) = ((aa) ∪ b)
9 oridm 110 . . . . . 6 (aa) = a
109ax-r5 38 . . . . 5 ((aa) ∪ b) = (ab)
118, 10ax-r2 36 . . . 4 ((ab) ∪ a) = (ab)
12 ancom 74 . . . 4 (a ∩ (ab)) = ((ab) ∩ a )
1311, 122or 72 . . 3 (((ab) ∪ a) ∪ (a ∩ (ab))) = ((ab) ∪ ((ab) ∩ a ))
14 orabs 120 . . 3 ((ab) ∪ ((ab) ∩ a )) = (ab)
1513, 14ax-r2 36 . 2 (((ab) ∪ a) ∪ (a ∩ (ab))) = (ab)
164, 2, 153tr2 64 1 ((a ∪ (a ∩ (ab))) ∪ (ab)) = (ab)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42 This theorem is referenced by:  woml  211  oml  445
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