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Theorem l42modlem1 1147
Description: Lemma for l42mod 1149..
Assertion
Ref Expression
l42modlem1 (((a v b) v d) ^ ((a v b) v e)) = ((a v b) v ((a v d) ^ (b v e)))
Proof of Theorem l42modlem1
StepHypRef Expression
1 leo 158 . . . . . 6 b =< (b v e)
21ml2i 1123 . . . . 5 (b v ((a v d) ^ (b v e))) = ((b v (a v d)) ^ (b v e))
3 ancom 74 . . . . 5 ((b v (a v d)) ^ (b v e)) = ((b v e) ^ (b v (a v d)))
42, 3tr 62 . . . 4 (b v ((a v d) ^ (b v e))) = ((b v e) ^ (b v (a v d)))
54lor 70 . . 3 (a v (b v ((a v d) ^ (b v e)))) = (a v ((b v e) ^ (b v (a v d))))
65cm 61 . 2 (a v ((b v e) ^ (b v (a v d)))) = (a v (b v ((a v d) ^ (b v e))))
7 orass 75 . . . . 5 ((a v b) v d) = (a v (b v d))
8 or12 80 . . . . 5 (a v (b v d)) = (b v (a v d))
97, 8tr 62 . . . 4 ((a v b) v d) = (b v (a v d))
10 orass 75 . . . 4 ((a v b) v e) = (a v (b v e))
119, 102an 79 . . 3 (((a v b) v d) ^ ((a v b) v e)) = ((b v (a v d)) ^ (a v (b v e)))
12 ancom 74 . . 3 ((b v (a v d)) ^ (a v (b v e))) = ((a v (b v e)) ^ (b v (a v d)))
13 leo 158 . . . . . 6 a =< (a v d)
1413lerr 150 . . . . 5 a =< (b v (a v d))
1514ml2i 1123 . . . 4 (a v ((b v e) ^ (b v (a v d)))) = ((a v (b v e)) ^ (b v (a v d)))
1615cm 61 . . 3 ((a v (b v e)) ^ (b v (a v d))) = (a v ((b v e) ^ (b v (a v d))))
1711, 12, 163tr 65 . 2 (((a v b) v d) ^ ((a v b) v e)) = (a v ((b v e) ^ (b v (a v d))))
18 orass 75 . 2 ((a v b) v ((a v d) ^ (b v e))) = (a v (b v ((a v d) ^ (b v e))))
196, 17, 183tr1 63 1 (((a v b) v d) ^ ((a v b) v e)) = ((a v b) v ((a v d) ^ (b v e)))
Colors of variables: term
Syntax hints:   = wb 1   v 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  ax-ml 1120
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:  l42mod  1149  testmod2  1213  testmod2expanded  1214
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