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Theorem wletr 396
 Description: Transitive law for l.e.
Hypotheses
Ref Expression
wletr.1 (a2 b) = 1
wletr.2 (b2 c) = 1
Assertion
Ref Expression
wletr (a2 c) = 1

Proof of Theorem wletr
StepHypRef Expression
1 wletr.1 . . . . . . . 8 (a2 b) = 1
21wdf-le2 379 . . . . . . 7 ((ab) ≡ b) = 1
32wr5-2v 366 . . . . . 6 (((ab) ∪ c) ≡ (bc)) = 1
43wr1 197 . . . . 5 ((bc) ≡ ((ab) ∪ c)) = 1
5 wletr.2 . . . . . 6 (b2 c) = 1
65wdf-le2 379 . . . . 5 ((bc) ≡ c) = 1
7 ax-a3 32 . . . . . 6 ((ab) ∪ c) = (a ∪ (bc))
87bi1 118 . . . . 5 (((ab) ∪ c) ≡ (a ∪ (bc))) = 1
94, 6, 8w3tr2 375 . . . 4 (c ≡ (a ∪ (bc))) = 1
109wlan 370 . . 3 ((ac) ≡ (a ∩ (a ∪ (bc)))) = 1
11 anabs 121 . . . 4 (a ∩ (a ∪ (bc))) = a
1211bi1 118 . . 3 ((a ∩ (a ∪ (bc))) ≡ a) = 1
1310, 12wr2 371 . 2 ((ac) ≡ a) = 1
1413wdf2le1 385 1 (a2 c) = 1
 Colors of variables: term Syntax hints:   = wb 1   ∪ 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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