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Theorem mli 1124
 Description: Inference version of modular law.
Hypothesis
Ref Expression
mli.1 ca
Assertion
Ref Expression
mli ((ab) ∪ c) = (a ∩ (bc))

Proof of Theorem mli
StepHypRef Expression
1 ancom 74 . . . 4 (ab) = (ba)
21ror 71 . . 3 ((ab) ∪ c) = ((ba) ∪ c)
3 orcom 73 . . 3 ((ba) ∪ c) = (c ∪ (ba))
4 mli.1 . . . 4 ca
54ml2i 1123 . . 3 (c ∪ (ba)) = ((cb) ∩ a)
62, 3, 53tr 65 . 2 ((ab) ∪ c) = ((cb) ∩ a)
7 orcom 73 . . 3 (cb) = (bc)
87ran 78 . 2 ((cb) ∩ a) = ((bc) ∩ a)
9 ancom 74 . 2 ((bc) ∩ a) = (a ∩ (bc))
106, 8, 93tr 65 1 ((ab) ∪ c) = (a ∩ (bc))
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2   ∪ 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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