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Theorem mlduali 1126
 Description: Inference version of dual of modular law.
Hypothesis
Ref Expression
mlduali.1 ac
Assertion
Ref Expression
mlduali ((ab) ∩ c) = (a ∪ (bc))

Proof of Theorem mlduali
StepHypRef Expression
1 ax-a2 31 . . . 4 (ab) = (ba)
21ran 78 . . 3 ((ab) ∩ c) = ((ba) ∩ c)
3 ancom 74 . . 3 ((ba) ∩ c) = (c ∩ (ba))
4 mlduali.1 . . . 4 ac
54mldual2i 1125 . . 3 (c ∩ (ba)) = ((cb) ∪ a)
62, 3, 53tr 65 . 2 ((ab) ∩ c) = ((cb) ∪ a)
7 ancom 74 . . 3 (cb) = (bc)
87ror 71 . 2 ((cb) ∪ a) = ((bc) ∪ a)
9 orcom 73 . 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:  ml3le  1127  modexp  1150  dp15lema  1152  dp35leme  1171  xdp15  1197  xxdp15  1200  xdp45lem  1202  xdp43lem  1203  xdp45  1204  xdp43  1205  3dp43  1206  testmod2  1213  testmod2expanded  1214
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