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Theorem mi 125
 Description: Mittelstaedt implication.
Assertion
Ref Expression
mi ((ab) ≡ b) = (b ∪ (ab ))

Proof of Theorem mi
StepHypRef Expression
1 dfb 94 . 2 ((ab) ≡ b) = (((ab) ∩ b) ∪ ((ab)b ))
2 ancom 74 . . . 4 ((ab) ∩ b) = (b ∩ (ab))
3 ax-a2 31 . . . . . 6 (ab) = (ba)
43lan 77 . . . . 5 (b ∩ (ab)) = (b ∩ (ba))
5 anabs 121 . . . . 5 (b ∩ (ba)) = b
64, 5ax-r2 36 . . . 4 (b ∩ (ab)) = b
72, 6ax-r2 36 . . 3 ((ab) ∩ b) = b
8 oran 87 . . . . . . 7 (ab) = (ab )
98con2 67 . . . . . 6 (ab) = (ab )
109ran 78 . . . . 5 ((ab)b ) = ((ab ) ∩ b )
11 anass 76 . . . . 5 ((ab ) ∩ b ) = (a ∩ (bb ))
1210, 11ax-r2 36 . . . 4 ((ab)b ) = (a ∩ (bb ))
13 anidm 111 . . . . 5 (bb ) = b
1413lan 77 . . . 4 (a ∩ (bb )) = (ab )
1512, 14ax-r2 36 . . 3 ((ab)b ) = (ab )
167, 152or 72 . 2 (((ab) ∩ b) ∪ ((ab)b )) = (b ∪ (ab ))
171, 16ax-r2 36 1 ((ab) ≡ b) = (b ∪ (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:  di  126  lei2  346
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