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Theorem 3vth4 807
Description: A 3-variable theorem.
Assertion
Ref Expression
3vth4 ((a ->2 b)' ->2 (b v c)) = ((a ->2 c)' ->2 (b v c))
Proof of Theorem 3vth4
StepHypRef Expression
1 3vth2 805 . . . 4 ((a ->2 b) ^ (b v c)') = ((a ->2 c) ^ (b v c)')
2 ax-a1 30 . . . . 5 (a ->2 b) = (a ->2 b)''
32ran 78 . . . 4 ((a ->2 b) ^ (b v c)') = ((a ->2 b)'' ^ (b v c)')
4 ax-a1 30 . . . . 5 (a ->2 c) = (a ->2 c)''
54ran 78 . . . 4 ((a ->2 c) ^ (b v c)') = ((a ->2 c)'' ^ (b v c)')
61, 3, 53tr2 64 . . 3 ((a ->2 b)'' ^ (b v c)') = ((a ->2 c)'' ^ (b v c)')
76lor 70 . 2 ((b v c) v ((a ->2 b)'' ^ (b v c)')) = ((b v c) v ((a ->2 c)'' ^ (b v c)'))
8 df-i2 45 . 2 ((a ->2 b)' ->2 (b v c)) = ((b v c) v ((a ->2 b)'' ^ (b v c)'))
9 df-i2 45 . 2 ((a ->2 c)' ->2 (b v c)) = ((b v c) v ((a ->2 c)'' ^ (b v c)'))
107, 8, 93tr1 63 1 ((a ->2 b)' ->2 (b v c)) = ((a ->2 c)' ->2 (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:  3vth6  809
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