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Theorem ml3le 1127
Description: Form of modular law that swaps two terms.
Assertion
Ref Expression
ml3le (a v (b ^ (c v a))) =< (a v (c ^ (b v a)))
Proof of Theorem ml3le
StepHypRef Expression
1 lear 161 . . . . 5 (b ^ (c v a)) =< (c v a)
21lelor 166 . . . 4 (a v (b ^ (c v a))) =< (a v (c v a))
3 or12 80 . . . . 5 (a v (c v a)) = (c v (a v a))
4 oridm 110 . . . . . 6 (a v a) = a
54lor 70 . . . . 5 (c v (a v a)) = (c v a)
6 orcom 73 . . . . 5 (c v a) = (a v c)
73, 5, 63tr 65 . . . 4 (a v (c v a)) = (a v c)
82, 7lbtr 139 . . 3 (a v (b ^ (c v a))) =< (a v c)
9 leor 159 . . . 4 a =< (b v a)
10 leao1 162 . . . 4 (b ^ (c v a)) =< (b v a)
119, 10lel2or 170 . . 3 (a v (b ^ (c v a))) =< (b v a)
128, 11ler2an 173 . 2 (a v (b ^ (c v a))) =< ((a v c) ^ (b v a))
139mlduali 1126 . 2 ((a v c) ^ (b v a)) = (a v (c ^ (b v a)))
1412, 13lbtr 139 1 (a v (b ^ (c v a))) =< (a v (c ^ (b v a)))
Colors of variables: term
Syntax hints:   =< 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:  ml3  1128  dp15leme  1156  xdp15  1197  xxdp15  1200  xdp45lem  1202  xdp43lem  1203  xdp45  1204  xdp43  1205  3dp43  1206
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