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Theorem lem3.3.6 1056
 Description: Equation 3.6 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton, 28-Jun-2005.) (Revised by Roy F. Longton, 3-Jul-2005.)
Assertion
Ref Expression
lem3.3.6 (a2 (bc)) = ((ac) →2 (bc))

Proof of Theorem lem3.3.6
StepHypRef Expression
1 anor3 90 . . . . . 6 (bc ) = (bc)
21ax-r1 35 . . . . 5 (bc) = (bc )
32lan 77 . . . 4 (a ∩ (bc) ) = (a ∩ (bc ))
4 anandir 115 . . . . 5 ((ab ) ∩ c ) = ((ac ) ∩ (bc ))
5 anass 76 . . . . 5 ((ab ) ∩ c ) = (a ∩ (bc ))
6 anor3 90 . . . . . 6 (ac ) = (ac)
76, 12an 79 . . . . 5 ((ac ) ∩ (bc )) = ((ac) ∩ (bc) )
84, 5, 73tr2 64 . . . 4 (a ∩ (bc )) = ((ac) ∩ (bc) )
93, 8ax-r2 36 . . 3 (a ∩ (bc) ) = ((ac) ∩ (bc) )
109lor 70 . 2 ((bc) ∪ (a ∩ (bc) )) = ((bc) ∪ ((ac) ∩ (bc) ))
11 df-i2 45 . 2 (a2 (bc)) = ((bc) ∪ (a ∩ (bc) ))
12 df-i2 45 . 2 ((ac) →2 (bc)) = ((bc) ∪ ((ac) ∩ (bc) ))
1310, 11, 123tr1 63 1 (a2 (bc)) = ((ac) →2 (bc))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →2 wi2 13 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-i2 45 This theorem is referenced by:  lem3.4.6  1079
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