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Theorem mccune2 247
 Description: E2 - OL theorem proved by EQP
Assertion
Ref Expression
mccune2 (a ∪ ((a ∩ ((ab ) ∩ (ab))) ∪ (a ∩ ((ab) ∪ (ab ))))) = 1

Proof of Theorem mccune2
StepHypRef Expression
1 ax-a3 32 . . 3 ((a ∪ ((ab ) ∩ (ab)) ) ∪ (a ∪ ((ab ) ∩ (ab)) ) ) = (a ∪ (((ab ) ∩ (ab)) ∪ (a ∪ ((ab ) ∩ (ab)) ) ))
21ax-r1 35 . 2 (a ∪ (((ab ) ∩ (ab)) ∪ (a ∪ ((ab ) ∩ (ab)) ) )) = ((a ∪ ((ab ) ∩ (ab)) ) ∪ (a ∪ ((ab ) ∩ (ab)) ) )
3 anor2 89 . . . . 5 (a ∩ ((ab ) ∩ (ab))) = (a ∪ ((ab ) ∩ (ab)) )
4 lear 161 . . . . . . 7 (a ∩ ((ab) ∪ (ab ))) ≤ ((ab) ∪ (ab ))
5 lea 160 . . . . . . . . 9 (ab) ≤ a
6 lea 160 . . . . . . . . 9 (ab ) ≤ a
75, 6lel2or 170 . . . . . . . 8 ((ab) ∪ (ab )) ≤ a
8 id 59 . . . . . . . . 9 ((ab) ∪ (ab )) = ((ab) ∪ (ab ))
98bile 142 . . . . . . . 8 ((ab) ∪ (ab )) ≤ ((ab) ∪ (ab ))
107, 9ler2an 173 . . . . . . 7 ((ab) ∪ (ab )) ≤ (a ∩ ((ab) ∪ (ab )))
114, 10lebi 145 . . . . . 6 (a ∩ ((ab) ∪ (ab ))) = ((ab) ∪ (ab ))
12 anor2 89 . . . . . . . 8 (ab) = (ab )
13 anor3 90 . . . . . . . 8 (ab ) = (ab)
1412, 132or 72 . . . . . . 7 ((ab) ∪ (ab )) = ((ab ) ∪ (ab) )
15 oran3 93 . . . . . . 7 ((ab ) ∪ (ab) ) = ((ab ) ∩ (ab))
1614, 15ax-r2 36 . . . . . 6 ((ab) ∪ (ab )) = ((ab ) ∩ (ab))
1711, 16ax-r2 36 . . . . 5 (a ∩ ((ab) ∪ (ab ))) = ((ab ) ∩ (ab))
183, 172or 72 . . . 4 ((a ∩ ((ab ) ∩ (ab))) ∪ (a ∩ ((ab) ∪ (ab )))) = ((a ∪ ((ab ) ∩ (ab)) ) ∪ ((ab ) ∩ (ab)) )
19 ax-a2 31 . . . 4 ((a ∪ ((ab ) ∩ (ab)) ) ∪ ((ab ) ∩ (ab)) ) = (((ab ) ∩ (ab)) ∪ (a ∪ ((ab ) ∩ (ab)) ) )
2018, 19ax-r2 36 . . 3 ((a ∩ ((ab ) ∩ (ab))) ∪ (a ∩ ((ab) ∪ (ab )))) = (((ab ) ∩ (ab)) ∪ (a ∪ ((ab ) ∩ (ab)) ) )
2120lor 70 . 2 (a ∪ ((a ∩ ((ab ) ∩ (ab))) ∪ (a ∩ ((ab) ∪ (ab ))))) = (a ∪ (((ab ) ∩ (ab)) ∪ (a ∪ ((ab ) ∩ (ab)) ) ))
22 df-t 41 . 2 1 = ((a ∪ ((ab ) ∩ (ab)) ) ∪ (a ∪ ((ab ) ∩ (ab)) ) )
232, 21, 223tr1 63 1 (a ∪ ((a ∩ ((ab ) ∩ (ab))) ∪ (a ∩ ((ab) ∪ (ab ))))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8 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  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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