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Theorem dp15leme 1156
Description: Part of proof (1)=>(5) in Day/Pickering 1982.
Hypotheses
Ref Expression
dp15lema.1 d = (a2 v (a0 ^ (a1 v b1)))
dp15lema.2 p0 = ((a1 v b1) ^ (a2 v b2))
dp15lema.3 e = (b0 ^ (a0 v p0))
Assertion
Ref Expression
dp15leme (((a0 v a2) ^ ((b0 ^ (a0 v p0)) v b2)) v (((a1 v a2) v (a0 ^ (a1 v b1))) ^ (b1 v b2))) =< (((a0 v a2) ^ ((b0 ^ (a0 v p0)) v b2)) v (((a1 v a2) v (b1 ^ (a0 v a1))) ^ (b1 v b2)))

Proof of Theorem dp15leme
StepHypRef Expression
1 ax-a2 31 . . . . . . 7 (a1 v a2) = (a2 v a1)
2 ax-a2 31 . . . . . . . 8 (a1 v b1) = (b1 v a1)
32lan 77 . . . . . . 7 (a0 ^ (a1 v b1)) = (a0 ^ (b1 v a1))
41, 32or 72 . . . . . 6 ((a1 v a2) v (a0 ^ (a1 v b1))) = ((a2 v a1) v (a0 ^ (b1 v a1)))
5 orass 75 . . . . . 6 ((a2 v a1) v (a0 ^ (b1 v a1))) = (a2 v (a1 v (a0 ^ (b1 v a1))))
64, 5tr 62 . . . . 5 ((a1 v a2) v (a0 ^ (a1 v b1))) = (a2 v (a1 v (a0 ^ (b1 v a1))))
7 ml3le 1127 . . . . . 6 (a1 v (a0 ^ (b1 v a1))) =< (a1 v (b1 ^ (a0 v a1)))
87lelor 166 . . . . 5 (a2 v (a1 v (a0 ^ (b1 v a1)))) =< (a2 v (a1 v (b1 ^ (a0 v a1))))
96, 8bltr 138 . . . 4 ((a1 v a2) v (a0 ^ (a1 v b1))) =< (a2 v (a1 v (b1 ^ (a0 v a1))))
10 orass 75 . . . . . 6 ((a2 v a1) v (b1 ^ (a0 v a1))) = (a2 v (a1 v (b1 ^ (a0 v a1))))
1110cm 61 . . . . 5 (a2 v (a1 v (b1 ^ (a0 v a1)))) = ((a2 v a1) v (b1 ^ (a0 v a1)))
12 ax-a2 31 . . . . . 6 (a2 v a1) = (a1 v a2)
1312ror 71 . . . . 5 ((a2 v a1) v (b1 ^ (a0 v a1))) = ((a1 v a2) v (b1 ^ (a0 v a1)))
1411, 13tr 62 . . . 4 (a2 v (a1 v (b1 ^ (a0 v a1)))) = ((a1 v a2) v (b1 ^ (a0 v a1)))
159, 14lbtr 139 . . 3 ((a1 v a2) v (a0 ^ (a1 v b1))) =< ((a1 v a2) v (b1 ^ (a0 v a1)))
1615leran 153 . 2 (((a1 v a2) v (a0 ^ (a1 v b1))) ^ (b1 v b2)) =< (((a1 v a2) v (b1 ^ (a0 v a1))) ^ (b1 v b2))
1716lelor 166 1 (((a0 v a2) ^ ((b0 ^ (a0 v p0)) v b2)) v (((a1 v a2) v (a0 ^ (a1 v b1))) ^ (b1 v b2))) =< (((a0 v a2) ^ ((b0 ^ (a0 v p0)) v b2)) v (((a1 v a2) v (b1 ^ (a0 v a1))) ^ (b1 v b2)))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2   v wo 6   ^ wa 7
This theorem is referenced by:  dp15lemh  1159
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
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