Theorem ud5lem0b 257

Description: Introduce ->5 to the right.

Hypothesis

Ref	Expression
ud5lem0a.1	a = b

Assertion

Ref	Expression
ud5lem0b	(a ->5 c) = (b ->5 c)

Proof of Theorem ud5lem0b

Step	Hyp	Ref	Expression
1		ud5lem0a.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	ran 71	. . 3 (a_|_ ^ c_|_) = (b_|_ ^ c_|_)
7	5, 6	2or 67	. 2 (((a ^ c) v (a_|_ ^ c)) v (a_|_ ^ c_|_)) = (((b ^ c) v (b_|_ ^ c)) v (b_|_ ^ c_|_))
8		df-i5 47	. 2 (a ->5 c) = (((a ^ c) v (a_|_ ^ c)) v (a_|_ ^ c_|_))
9		df-i5 47	. 2 (b ->5 c) = (((b ^ c) v (b_|_ ^ c)) v (b_|_ ^ c_|_))
10	7, 8, 9	3tr1 60	1 (a ->5 c) = (b ->5 c)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->5 wi5 17

This theorem is referenced by:  ud5 581

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-i5 47

metamath.org