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Theorem mli 1124
Description: Inference version of modular law.
Hypothesis
Ref Expression
mli.1 c =< a
Assertion
Ref Expression
mli ((a ^ b) v c) = (a ^ (b v c))
Proof of Theorem mli
StepHypRef Expression
1 ancom 74 . . . 4 (a ^ b) = (b ^ a)
21ror 71 . . 3 ((a ^ b) v c) = ((b ^ a) v c)
3 orcom 73 . . 3 ((b ^ a) v c) = (c v (b ^ a))
4 mli.1 . . . 4 c =< a
54ml2i 1123 . . 3 (c v (b ^ a)) = ((c v b) ^ a)
62, 3, 53tr 65 . 2 ((a ^ b) v c) = ((c v b) ^ a)
7 orcom 73 . . 3 (c v b) = (b v c)
87ran 78 . 2 ((c v b) ^ a) = ((b v c) ^ a)
9 ancom 74 . 2 ((b v c) ^ a) = (a ^ (b v c))
106, 8, 93tr 65 1 ((a ^ b) v c) = (a ^ (b v c))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2   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:  dp41lemf  1186  xdp41  1196  xxdp41  1199  xdp45lem  1202  xdp43lem  1203  xdp45  1204  xdp43  1205  3dp43  1206  testmod  1211  testmod3  1215
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