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Theorem mldual 1122
Description: Dual of modular law.
Assertion
Ref Expression
mldual (a ^ (b v (a ^ c))) = ((a ^ b) v (a ^ c))
Proof of Theorem mldual
StepHypRef Expression
1 anor3 90 . . . . . . 7 (b' ^ (a ^ c)') = (b v (a ^ c))'
21cm 61 . . . . . 6 (b v (a ^ c))' = (b' ^ (a ^ c)')
3 oran3 93 . . . . . . . 8 (a' v c') = (a ^ c)'
43lan 77 . . . . . . 7 (b' ^ (a' v c')) = (b' ^ (a ^ c)')
54ax-r1 35 . . . . . 6 (b' ^ (a ^ c)') = (b' ^ (a' v c'))
62, 5tr 62 . . . . 5 (b v (a ^ c))' = (b' ^ (a' v c'))
76lor 70 . . . 4 (a' v (b v (a ^ c))') = (a' v (b' ^ (a' v c')))
8 ml 1121 . . . 4 (a' v (b' ^ (a' v c'))) = ((a' v b') ^ (a' v c'))
9 oran3 93 . . . . 5 (a' v b') = (a ^ b)'
109, 32an 79 . . . 4 ((a' v b') ^ (a' v c')) = ((a ^ b)' ^ (a ^ c)')
117, 8, 103tr 65 . . 3 (a' v (b v (a ^ c))') = ((a ^ b)' ^ (a ^ c)')
12 oran3 93 . . 3 (a' v (b v (a ^ c))') = (a ^ (b v (a ^ c)))'
13 anor3 90 . . 3 ((a ^ b)' ^ (a ^ c)') = ((a ^ b) v (a ^ c))'
1411, 12, 133tr2 64 . 2 (a ^ (b v (a ^ c)))' = ((a ^ b) v (a ^ c))'
1514con1 66 1 (a ^ (b v (a ^ c))) = ((a ^ b) v (a ^ c))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   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:  mldual2i  1125  vneulem13  1141  vneulemexp  1146  dp41lemd  1184  dp41leme  1185  dp32  1194  xdp41  1196  xxdp41  1199  xdp45lem  1202  xdp43lem  1203  xdp45  1204  xdp43  1205  3dp43  1206
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