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Theorem 3vth2 805
 Description: A 3-variable theorem. Equivalent to OML.
Assertion
Ref Expression
3vth2 ((a2 b) ∩ (bc) ) = ((a2 c) ∩ (bc) )

Proof of Theorem 3vth2
StepHypRef Expression
1 3vth1 804 . . 3 ((a2 b) ∩ (bc) ) ≤ (a2 c)
2 lear 161 . . 3 ((a2 b) ∩ (bc) ) ≤ (bc)
31, 2ler2an 173 . 2 ((a2 b) ∩ (bc) ) ≤ ((a2 c) ∩ (bc) )
4 ax-a2 31 . . . . . 6 (bc) = (cb)
54ax-r4 37 . . . . 5 (bc) = (cb)
65lan 77 . . . 4 ((a2 c) ∩ (bc) ) = ((a2 c) ∩ (cb) )
7 3vth1 804 . . . 4 ((a2 c) ∩ (cb) ) ≤ (a2 b)
86, 7bltr 138 . . 3 ((a2 c) ∩ (bc) ) ≤ (a2 b)
9 lear 161 . . 3 ((a2 c) ∩ (bc) ) ≤ (bc)
108, 9ler2an 173 . 2 ((a2 c) ∩ (bc) ) ≤ ((a2 b) ∩ (bc) )
113, 10lebi 145 1 ((a2 b) ∩ (bc) ) = ((a2 c) ∩ (bc) )
 Colors of variables: term Syntax hints:   = wb 1  ⊥ 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:  3vth4  807
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