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Theorem ml 1123
 Description: Modular law in equational form. (Contributed by NM, 15-Mar-2010.) (Revised by NM, 31-Mar-2011.)
Assertion
Ref Expression
ml (a ∪ (b ∩ (ac))) = ((ab) ∩ (ac))

Proof of Theorem ml
StepHypRef Expression
1 leo 158 . . . 4 a ≤ (ab)
2 leo 158 . . . 4 a ≤ (ac)
31, 2ler2an 173 . . 3 a ≤ ((ab) ∩ (ac))
4 leor 159 . . . 4 b ≤ (ab)
54leran 153 . . 3 (b ∩ (ac)) ≤ ((ab) ∩ (ac))
63, 5lel2or 170 . 2 (a ∪ (b ∩ (ac))) ≤ ((ab) ∩ (ac))
7 ax-ml 1122 . 2 ((ab) ∩ (ac)) ≤ (a ∪ (b ∩ (ac)))
86, 7lebi 145 1 (a ∪ (b ∩ (ac))) = ((ab) ∩ (ac))
 Colors of variables: term Syntax hints:   = wb 1   ∪ 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:  mldual  1124  ml2i  1125  vneulem2  1132  vneulem5  1135  vneulem12  1142  vneulemexp  1148
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