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

Step	Hyp	Ref	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)''
3	2	ran 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)''
5	4	ran 78	. . . 4 ((a ->2 c) ^ (b v c)') = ((a ->2 c)'' ^ (b v c)')
6	1, 3, 5	3tr2 64	. . 3 ((a ->2 b)'' ^ (b v c)') = ((a ->2 c)'' ^ (b v c)')
7	6	lor 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)'))
10	7, 8, 9	3tr1 63	1 ((a ->2 b)' ->2 (b v c)) = ((a ->2 c)' ->2 (b v c))

Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->2 wi2 13

This theorem is referenced by:  3vth6 809

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

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