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Theorem 3vth1 804
 Description: A 3-variable theorem. Equivalent to OML.
Assertion
Ref Expression
3vth1 ((a2 b) ∩ (bc) ) ≤ (a2 c)

Proof of Theorem 3vth1
StepHypRef Expression
1 anor3 90 . . . . . . 7 (bc ) = (bc)
21lan 77 . . . . . 6 ((b ∪ (ba )) ∩ (bc )) = ((b ∪ (ba )) ∩ (bc) )
32ax-r1 35 . . . . 5 ((b ∪ (ba )) ∩ (bc) ) = ((b ∪ (ba )) ∩ (bc ))
4 anass 76 . . . . . 6 (((b ∪ (ba )) ∩ b ) ∩ c ) = ((b ∪ (ba )) ∩ (bc ))
54ax-r1 35 . . . . 5 ((b ∪ (ba )) ∩ (bc )) = (((b ∪ (ba )) ∩ b ) ∩ c )
63, 5ax-r2 36 . . . 4 ((b ∪ (ba )) ∩ (bc) ) = (((b ∪ (ba )) ∩ b ) ∩ c )
7 ancom 74 . . . . . . 7 ((b ∪ (ba )) ∩ b ) = (b ∩ (b ∪ (ba )))
8 omlan 448 . . . . . . 7 (b ∩ (b ∪ (ba ))) = (ba )
97, 8ax-r2 36 . . . . . 6 ((b ∪ (ba )) ∩ b ) = (ba )
10 lear 161 . . . . . 6 (ba ) ≤ a
119, 10bltr 138 . . . . 5 ((b ∪ (ba )) ∩ b ) ≤ a
1211leran 153 . . . 4 (((b ∪ (ba )) ∩ b ) ∩ c ) ≤ (ac )
136, 12bltr 138 . . 3 ((b ∪ (ba )) ∩ (bc) ) ≤ (ac )
14 leor 159 . . 3 (ac ) ≤ (c ∪ (ac ))
1513, 14letr 137 . 2 ((b ∪ (ba )) ∩ (bc) ) ≤ (c ∪ (ac ))
16 df-i2 45 . . . 4 (a2 b) = (b ∪ (ab ))
17 ancom 74 . . . . 5 (ab ) = (ba )
1817lor 70 . . . 4 (b ∪ (ab )) = (b ∪ (ba ))
1916, 18ax-r2 36 . . 3 (a2 b) = (b ∪ (ba ))
2019ran 78 . 2 ((a2 b) ∩ (bc) ) = ((b ∪ (ba )) ∩ (bc) )
21 df-i2 45 . 2 (a2 c) = (c ∪ (ac ))
2215, 20, 21le3tr1 140 1 ((a2 b) ∩ (bc) ) ≤ (a2 c)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →2 wi2 13 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-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by:  3vth2  805  3vth3  806
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