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Theorem k1-8b 356
 Description: Second part of statement (8) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21. (Contributed by NM, 27-May-2008.)
Hypotheses
Ref Expression
k1-8b.1 y = ((yc) ∪ (yc ))
k1-8b.2 xc
k1-8b.3 yc
Assertion
Ref Expression
k1-8b y = ((xy) ∩ c )

Proof of Theorem k1-8b
StepHypRef Expression
1 k1-8b.1 . . . 4 y = ((yc) ∪ (yc ))
2 ax-a1 30 . . . . . 6 c = c
32lan 77 . . . . 5 (yc) = (yc )
43ror 71 . . . 4 ((yc) ∪ (yc )) = ((yc ) ∪ (yc ))
5 orcom 73 . . . 4 ((yc ) ∪ (yc )) = ((yc ) ∪ (yc ))
61, 4, 53tr 65 . . 3 y = ((yc ) ∪ (yc ))
7 k1-8b.3 . . 3 yc
8 k1-8b.2 . . . 4 xc
98, 2lbtr 139 . . 3 xc
106, 7, 9k1-8a 355 . 2 y = ((yx) ∩ c )
11 orcom 73 . . 3 (yx) = (xy)
1211ran 78 . 2 ((yx) ∩ c ) = ((xy) ∩ c )
1310, 12tr 62 1 y = ((xy) ∩ c )
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ 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-t 41  df-f 42  df-le1 130  df-le2 131 This theorem is referenced by:  k1-3  358
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