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Theorem wcom3i 422
 Description: Lemma 3(i) of Kalmbach 83 p. 23.
Hypothesis
Ref Expression
wcom3i.1 ((a ∩ (ab)) ≡ (ab)) = 1
Assertion
Ref Expression
wcom3i C (a, b) = 1

Proof of Theorem wcom3i
StepHypRef Expression
1 anor1 88 . . . . . . . . 9 (ab ) = (ab)
21bi1 118 . . . . . . . 8 ((ab ) ≡ (ab) ) = 1
32wcon2 208 . . . . . . 7 ((ab ) ≡ (ab)) = 1
43wran 369 . . . . . 6 (((ab )a) ≡ ((ab) ∩ a)) = 1
5 ancom 74 . . . . . . 7 ((ab) ∩ a) = (a ∩ (ab))
65bi1 118 . . . . . 6 (((ab) ∩ a) ≡ (a ∩ (ab))) = 1
74, 6wr2 371 . . . . 5 (((ab )a) ≡ (a ∩ (ab))) = 1
8 wcom3i.1 . . . . 5 ((a ∩ (ab)) ≡ (ab)) = 1
97, 8wr2 371 . . . 4 (((ab )a) ≡ (ab)) = 1
109wlor 368 . . 3 (((ab ) ∪ ((ab )a)) ≡ ((ab ) ∪ (ab))) = 1
11 wlea 388 . . . 4 ((ab ) ≤2 a) = 1
1211wom4 380 . . 3 (((ab ) ∪ ((ab )a)) ≡ a) = 1
13 ax-a2 31 . . . 4 ((ab ) ∪ (ab)) = ((ab) ∪ (ab ))
1413bi1 118 . . 3 (((ab ) ∪ (ab)) ≡ ((ab) ∪ (ab ))) = 1
1510, 12, 14w3tr2 375 . 2 (a ≡ ((ab) ∪ (ab ))) = 1
1615wdf-c1 383 1 C (a, b) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   C wcmtr 29 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-wom 361 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134 This theorem is referenced by: (None)
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