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

Theorem di 126
 Description: Dishkant implication.
Assertion
Ref Expression
di ((ab) ≡ a) = (a ∪ (ab))

Proof of Theorem di
StepHypRef Expression
1 conb 122 . . 3 ((ba )a) = ((ba ) a )
2 ax-a1 30 . . . . . 6 (ba ) = (ba )
32ax-r1 35 . . . . 5 (ba ) = (ba )
43rbi 98 . . . 4 ((ba ) a ) = ((ba ) ≡ a )
5 mi 125 . . . 4 ((ba ) ≡ a ) = (a ∪ (b a ))
64, 5ax-r2 36 . . 3 ((ba ) a ) = (a ∪ (b a ))
71, 6ax-r2 36 . 2 ((ba )a) = (a ∪ (b a ))
8 ancom 74 . . . 4 (ab) = (ba)
9 df-a 40 . . . 4 (ba) = (ba )
108, 9ax-r2 36 . . 3 (ab) = (ba )
1110rbi 98 . 2 ((ab) ≡ a) = ((ba )a)
12 ax-a1 30 . . . . 5 b = b
13 ax-a1 30 . . . . 5 a = a
1412, 132an 79 . . . 4 (ba) = (b a )
158, 14ax-r2 36 . . 3 (ab) = (b a )
1615lor 70 . 2 (a ∪ (ab)) = (a ∪ (b a ))
177, 11, 163tr1 63 1 ((ab) ≡ a) = (a ∪ (ab))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ 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-b 39  df-a 40  df-t 41  df-f 42 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator