Theorem ri3 245

Description: Introduce Kalmbach implication to the right.

Hypothesis

Ref	Expression
ri3.1	a = b

Assertion

Ref	Expression
ri3	(a ->3 c) = (b ->3 c)

Proof of Theorem ri3

Step	Hyp	Ref	Expression
1		ri3.1	. . . . . 6 a = b
2	1	ax-r4 36	. . . . 5 a_|_ = b_|_
3	2	ran 71	. . . 4 (a_|_ ^ c) = (b_|_ ^ c)
4	2	ran 71	. . . 4 (a_|_ ^ c_|_) = (b_|_ ^ c_|_)
5	3, 4	2or 67	. . 3 ((a_|_ ^ c) v (a_|_ ^ c_|_)) = ((b_|_ ^ c) v (b_|_ ^ c_|_))
6	2	ax-r5 37	. . . 4 (a_|_ v c) = (b_|_ v c)
7	1, 6	2an 72	. . 3 (a ^ (a_|_ v c)) = (b ^ (b_|_ v c))
8	5, 7	2or 67	. 2 (((a_|_ ^ c) v (a_|_ ^ c_|_)) v (a ^ (a_|_ v c))) = (((b_|_ ^ c) v (b_|_ ^ c_|_)) v (b ^ (b_|_ v c)))
9		df-i3 45	. 2 (a ->3 c) = (((a_|_ ^ c) v (a_|_ ^ c_|_)) v (a ^ (a_|_ v c)))
10		df-i3 45	. 2 (b ->3 c) = (((b_|_ ^ c) v (b_|_ ^ c_|_)) v (b ^ (b_|_ v c)))
11	8, 9, 10	3tr1 60	1 (a ->3 c) = (b ->3 c)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->3 wi3 15

This theorem is referenced by:  2i3 246  ud3lem0b 253  bina2 275  ska14 496  i3orcom 507  i3ancom 508  bi3tr 509  i3ri3 520

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-i3 45

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