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Theorem u4lemob 633
 Description: Lemma for non-tollens implication study. (Contributed by NM, 15-Dec-1997.)
Assertion
Ref Expression
u4lemob ((a4 b) ∪ b) = (ab)

Proof of Theorem u4lemob
StepHypRef Expression
1 df-i4 47 . . 3 (a4 b) = (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))
21ax-r5 38 . 2 ((a4 b) ∪ b) = ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∪ b)
3 or32 82 . . 3 ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∪ b) = ((((ab) ∪ (ab)) ∪ b) ∪ ((ab) ∩ b ))
4 lear 161 . . . . . . 7 (ab) ≤ b
5 lear 161 . . . . . . 7 (ab) ≤ b
64, 5lel2or 170 . . . . . 6 ((ab) ∪ (ab)) ≤ b
76df-le2 131 . . . . 5 (((ab) ∪ (ab)) ∪ b) = b
87ax-r5 38 . . . 4 ((((ab) ∪ (ab)) ∪ b) ∪ ((ab) ∩ b )) = (b ∪ ((ab) ∩ b ))
9 comorr2 463 . . . . . 6 b C (ab)
10 comid 187 . . . . . . 7 b C b
1110comcom2 183 . . . . . 6 b C b
129, 11fh3 471 . . . . 5 (b ∪ ((ab) ∩ b )) = ((b ∪ (ab)) ∩ (bb ))
13 or12 80 . . . . . . . 8 (b ∪ (ab)) = (a ∪ (bb))
14 oridm 110 . . . . . . . . 9 (bb) = b
1514lor 70 . . . . . . . 8 (a ∪ (bb)) = (ab)
1613, 15ax-r2 36 . . . . . . 7 (b ∪ (ab)) = (ab)
17 df-t 41 . . . . . . . 8 1 = (bb )
1817ax-r1 35 . . . . . . 7 (bb ) = 1
1916, 182an 79 . . . . . 6 ((b ∪ (ab)) ∩ (bb )) = ((ab) ∩ 1)
20 an1 106 . . . . . 6 ((ab) ∩ 1) = (ab)
2119, 20ax-r2 36 . . . . 5 ((b ∪ (ab)) ∩ (bb )) = (ab)
2212, 21ax-r2 36 . . . 4 (b ∪ ((ab) ∩ b )) = (ab)
238, 22ax-r2 36 . . 3 ((((ab) ∪ (ab)) ∪ b) ∪ ((ab) ∩ b )) = (ab)
243, 23ax-r2 36 . 2 ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∪ b) = (ab)
252, 24ax-r2 36 1 ((a4 b) ∪ b) = (ab)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →4 wi4 15 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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u4lemnanb  658
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