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Theorem wwcom3ii 215
 Description: Lemma 3(ii) (weak) of Kalmbach 83 p. 23.
Hypothesis
Ref Expression
wwcom3ii.1 b C a
Assertion
Ref Expression
wwcom3ii (a ∩ (ab)) = (ab)

Proof of Theorem wwcom3ii
StepHypRef Expression
1 wwcom3ii.1 . . . . 5 b C a
21wwcomd 214 . . . 4 b = ((ba) ∩ (ba ))
32lan 77 . . 3 (ab) = (a ∩ ((ba) ∩ (ba )))
4 anass 76 . . . . 5 ((a ∩ (ba)) ∩ (ba )) = (a ∩ ((ba) ∩ (ba )))
54ax-r1 35 . . . 4 (a ∩ ((ba) ∩ (ba ))) = ((a ∩ (ba)) ∩ (ba ))
6 ax-a2 31 . . . . . . 7 (ba) = (ab)
76lan 77 . . . . . 6 (a ∩ (ba)) = (a ∩ (ab))
8 anabs 121 . . . . . 6 (a ∩ (ab)) = a
97, 8ax-r2 36 . . . . 5 (a ∩ (ba)) = a
10 ax-a2 31 . . . . 5 (ba ) = (ab)
119, 102an 79 . . . 4 ((a ∩ (ba)) ∩ (ba )) = (a ∩ (ab))
125, 11ax-r2 36 . . 3 (a ∩ ((ba) ∩ (ba ))) = (a ∩ (ab))
133, 12ax-r2 36 . 2 (ab) = (a ∩ (ab))
1413ax-r1 35 1 (a ∩ (ab)) = (ab)
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ∪ 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-a 40  df-c2 133 This theorem is referenced by:  wwfh1  216  wwfh2  217
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