Theorem ud4lem0b 255

Description: Introduce ->4 to the right.

Hypothesis

Ref	Expression
ud4lem0a.1	a = b

Assertion

Ref	Expression
ud4lem0b	(a ->4 c) = (b ->4 c)

Proof of Theorem ud4lem0b

Step	Hyp	Ref	Expression
1		ud4lem0a.1	. . . . 5 a = b
2	1	ran 71	. . . 4 (a ^ c) = (b ^ c)
3	1	ax-r4 36	. . . . 5 a_|_ = b_|_
4	3	ran 71	. . . 4 (a_|_ ^ c) = (b_|_ ^ c)
5	2, 4	2or 67	. . 3 ((a ^ c) v (a_|_ ^ c)) = ((b ^ c) v (b_|_ ^ c))
6	3	ax-r5 37	. . . 4 (a_|_ v c) = (b_|_ v c)
7	6	ran 71	. . 3 ((a_|_ v c) ^ c_|_) = ((b_|_ v c) ^ c_|_)
8	5, 7	2or 67	. 2 (((a ^ c) v (a_|_ ^ c)) v ((a_|_ v c) ^ c_|_)) = (((b ^ c) v (b_|_ ^ c)) v ((b_|_ v c) ^ c_|_))
9		df-i4 46	. 2 (a ->4 c) = (((a ^ c) v (a_|_ ^ c)) v ((a_|_ v c) ^ c_|_))
10		df-i4 46	. 2 (b ->4 c) = (((b ^ c) v (b_|_ ^ c)) v ((b_|_ v c) ^ c_|_))
11	8, 9, 10	3tr1 60	1 (a ->4 c) = (b ->4 c)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->4 wi4 16

This theorem is referenced by:  i4i3 263  ud4 580

This theorem was proved from axioms:  ax-a2 30  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-i4 46

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