Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  i2bi GIF version

Theorem i2bi 722
 Description: Dishkant implication expressed with biconditional.
Assertion
Ref Expression
i2bi (a2 b) = (b ∪ (ab))

Proof of Theorem i2bi
StepHypRef Expression
1 leor 159 . . . 4 (ab ) ≤ ((ab) ∪ (ab ))
21lelor 166 . . 3 (b ∪ (ab )) ≤ (b ∪ ((ab) ∪ (ab )))
3 df-i2 45 . . 3 (a2 b) = (b ∪ (ab ))
4 dfb 94 . . . 4 (ab) = ((ab) ∪ (ab ))
54lor 70 . . 3 (b ∪ (ab)) = (b ∪ ((ab) ∪ (ab )))
62, 3, 5le3tr1 140 . 2 (a2 b) ≤ (b ∪ (ab))
7 leo 158 . . . 4 b ≤ (b ∪ (ab ))
83ax-r1 35 . . . 4 (b ∪ (ab )) = (a2 b)
97, 8lbtr 139 . . 3 b ≤ (a2 b)
10 u2lembi 721 . . . . 5 ((a2 b) ∩ (b2 a)) = (ab)
1110ax-r1 35 . . . 4 (ab) = ((a2 b) ∩ (b2 a))
12 lea 160 . . . 4 ((a2 b) ∩ (b2 a)) ≤ (a2 b)
1311, 12bltr 138 . . 3 (ab) ≤ (a2 b)
149, 13lel2or 170 . 2 (b ∪ (ab)) ≤ (a2 b)
156, 14lebi 145 1 (a2 b) = (b ∪ (ab))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →2 wi2 13 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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  mloa  1018
 Copyright terms: Public domain W3C validator