Theorem mi 117

Description: Mittelstaedt implication.

Assertion

Ref	Expression
mi	((a v b) == b) = (b v (a_|_ ^ b_|_))

Proof of Theorem mi

Step	Hyp	Ref	Expression
1		dfb 86	. 2 ((a v b) == b) = (((a v b) ^ b) v ((a v b)_|_ ^ b_|_))
2		ancom 68	. . . 4 ((a v b) ^ b) = (b ^ (a v b))
3		ax-a2 30	. . . . . 6 (a v b) = (b v a)
4	3	lan 70	. . . . 5 (b ^ (a v b)) = (b ^ (b v a))
5		a5c 113	. . . . 5 (b ^ (b v a)) = b
6	4, 5	ax-r2 35	. . . 4 (b ^ (a v b)) = b
7	2, 6	ax-r2 35	. . 3 ((a v b) ^ b) = b
8		oran 79	. . . . . . 7 (a v b) = (a_|_ ^ b_|_)_|_
9	8	con2 64	. . . . . 6 (a v b)_|_ = (a_|_ ^ b_|_)
10	9	ran 71	. . . . 5 ((a v b)_|_ ^ b_|_) = ((a_|_ ^ b_|_) ^ b_|_)
11		anass 69	. . . . 5 ((a_|_ ^ b_|_) ^ b_|_) = (a_|_ ^ (b_|_ ^ b_|_))
12	10, 11	ax-r2 35	. . . 4 ((a v b)_|_ ^ b_|_) = (a_|_ ^ (b_|_ ^ b_|_))
13		anidm 103	. . . . 5 (b_|_ ^ b_|_) = b_|_
14	13	lan 70	. . . 4 (a_|_ ^ (b_|_ ^ b_|_)) = (a_|_ ^ b_|_)
15	12, 14	ax-r2 35	. . 3 ((a v b)_|_ ^ b_|_) = (a_|_ ^ b_|_)
16	7, 15	2or 67	. 2 (((a v b) ^ b) v ((a v b)_|_ ^ b_|_)) = (b v (a_|_ ^ b_|_))
17	1, 16	ax-r2 35	1 ((a v b) == b) = (b v (a_|_ ^ b_|_))

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7

This theorem is referenced by:  di 118  lei2 338

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

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41

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