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Theorem 3vth2 805
Description: A 3-variable theorem. Equivalent to OML.
Assertion
Ref Expression
3vth2 ((a ->2 b) ^ (b v c)') = ((a ->2 c) ^ (b v c)')
Proof of Theorem 3vth2
StepHypRef Expression
1 3vth1 804 . . 3 ((a ->2 b) ^ (b v c)') =< (a ->2 c)
2 lear 161 . . 3 ((a ->2 b) ^ (b v c)') =< (b v c)'
31, 2ler2an 173 . 2 ((a ->2 b) ^ (b v c)') =< ((a ->2 c) ^ (b v c)')
4 ax-a2 31 . . . . . 6 (b v c) = (c v b)
54ax-r4 37 . . . . 5 (b v c)' = (c v b)'
65lan 77 . . . 4 ((a ->2 c) ^ (b v c)') = ((a ->2 c) ^ (c v b)')
7 3vth1 804 . . . 4 ((a ->2 c) ^ (c v b)') =< (a ->2 b)
86, 7bltr 138 . . 3 ((a ->2 c) ^ (b v c)') =< (a ->2 b)
9 lear 161 . . 3 ((a ->2 c) ^ (b v c)') =< (b v c)'
108, 9ler2an 173 . 2 ((a ->2 c) ^ (b v c)') =< ((a ->2 b) ^ (b v c)')
113, 10lebi 145 1 ((a ->2 b) ^ (b v c)') = ((a ->2 c) ^ (b v c)')
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v 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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