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Theorem ml3le 1129
 Description: Form of modular law that swaps two terms. (Contributed by NM, 1-Apr-2012.)
Assertion
Ref Expression
ml3le (a ∪ (b ∩ (ca))) ≤ (a ∪ (c ∩ (ba)))

Proof of Theorem ml3le
StepHypRef Expression
1 lear 161 . . . . 5 (b ∩ (ca)) ≤ (ca)
21lelor 166 . . . 4 (a ∪ (b ∩ (ca))) ≤ (a ∪ (ca))
3 or12 80 . . . . 5 (a ∪ (ca)) = (c ∪ (aa))
4 oridm 110 . . . . . 6 (aa) = a
54lor 70 . . . . 5 (c ∪ (aa)) = (ca)
6 orcom 73 . . . . 5 (ca) = (ac)
73, 5, 63tr 65 . . . 4 (a ∪ (ca)) = (ac)
82, 7lbtr 139 . . 3 (a ∪ (b ∩ (ca))) ≤ (ac)
9 leor 159 . . . 4 a ≤ (ba)
10 leao1 162 . . . 4 (b ∩ (ca)) ≤ (ba)
119, 10lel2or 170 . . 3 (a ∪ (b ∩ (ca))) ≤ (ba)
128, 11ler2an 173 . 2 (a ∪ (b ∩ (ca))) ≤ ((ac) ∩ (ba))
139mlduali 1128 . 2 ((ac) ∩ (ba)) = (a ∪ (c ∩ (ba)))
1412, 13lbtr 139 1 (a ∪ (b ∩ (ca))) ≤ (a ∪ (c ∩ (ba)))
 Colors of variables: term Syntax hints:   ≤ 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 1122 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:  ml3  1130  dp15leme  1158  xdp15  1199  xxdp15  1202  xdp45lem  1204  xdp43lem  1205  xdp45  1206  xdp43  1207  3dp43  1208
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