Theorem 3vth4 789

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 787	. . . 4 ((a ->2 b) ^ (b v c)_|_) = ((a ->2 c) ^ (b v c)_|_)
2		ax-a1 29	. . . . 5 (a ->2 b) = (a ->2 b)_|__|_
3	2	ran 71	. . . 4 ((a ->2 b) ^ (b v c)_|_) = ((a ->2 b)_|__|_ ^ (b v c)_|_)
4		ax-a1 29	. . . . 5 (a ->2 c) = (a ->2 c)_|__|_
5	4	ran 71	. . . 4 ((a ->2 c) ^ (b v c)_|_) = ((a ->2 c)_|__|_ ^ (b v c)_|_)
6	1, 3, 5	3tr2 61	. . 3 ((a ->2 b)_|__|_ ^ (b v c)_|_) = ((a ->2 c)_|__|_ ^ (b v c)_|_)
7	6	lor 66	. 2 ((b v c) v ((a ->2 b)_|__|_ ^ (b v c)_|_)) = ((b v c) v ((a ->2 c)_|__|_ ^ (b v c)_|_))
8		df-i2 44	. 2 ((a ->2 b)_|_ ->2 (b v c)) = ((b v c) v ((a ->2 b)_|__|_ ^ (b v c)_|_))
9		df-i2 44	. 2 ((a ->2 c)_|_ ->2 (b v c)) = ((b v c) v ((a ->2 c)_|__|_ ^ (b v c)_|_))
10	7, 8, 9	3tr1 60	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 14

This theorem is referenced by:  3vth6 791

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i2 44  df-le1 122  df-le2 123

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