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Theorem wletr 396
Description: Transitive law for l.e.
Hypotheses
Ref Expression
wletr.1 (a =<2 b) = 1
wletr.2 (b =<2 c) = 1
Assertion
Ref Expression
wletr (a =<2 c) = 1
Proof of Theorem wletr
StepHypRef Expression
1 wletr.1 . . . . . . . 8 (a =<2 b) = 1
21wdf-le2 379 . . . . . . 7 ((a v b) == b) = 1
32wr5-2v 366 . . . . . 6 (((a v b) v c) == (b v c)) = 1
43wr1 197 . . . . 5 ((b v c) == ((a v b) v c)) = 1
5 wletr.2 . . . . . 6 (b =<2 c) = 1
65wdf-le2 379 . . . . 5 ((b v c) == c) = 1
7 ax-a3 32 . . . . . 6 ((a v b) v c) = (a v (b v c))
87bi1 118 . . . . 5 (((a v b) v c) == (a v (b v c))) = 1
94, 6, 8w3tr2 375 . . . 4 (c == (a v (b v c))) = 1
109wlan 370 . . 3 ((a ^ c) == (a ^ (a v (b v c)))) = 1
11 anabs 121 . . . 4 (a ^ (a v (b v c))) = a
1211bi1 118 . . 3 ((a ^ (a v (b v c))) == a) = 1
1310, 12wr2 371 . 2 ((a ^ c) == a) = 1
1413wdf2le1 385 1 (a =<2 c) = 1
Colors of variables: term
Syntax hints:   = wb 1   v wo 6   ^ wa 7  1wt 8   =<2 wle2 10
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-wom 361
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131
This theorem is referenced by:  wle2or  403  wle2an  404  ska4  433
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