Theorem biao 781

Description: Equivalence to biconditional.

Assertion

Ref	Expression
biao	(a == b) = ((a ^ b) == (a v b))

Proof of Theorem biao

Step	Hyp	Ref	Expression
1		leao1 154	. . . . 5 (a ^ b) =< (a v b)
2	1	df2le2 128	. . . 4 ((a ^ b) ^ (a v b)) = (a ^ b)
3	2	ax-r1 34	. . 3 (a ^ b) = ((a ^ b) ^ (a v b))
4		anor3 82	. . . 4 (a_|_ ^ b_|_) = (a v b)_|_
5	1	lecon 146	. . . . . 6 (a v b)_|_ =< (a ^ b)_|_
6		oridm 102	. . . . . . 7 ((a v b)_|_ v (a v b)_|_) = (a v b)_|_
7	6	df-le1 122	. . . . . 6 (a v b)_|_ =< (a v b)_|_
8	5, 7	ler2an 165	. . . . 5 (a v b)_|_ =< ((a ^ b)_|_ ^ (a v b)_|_)
9		lear 153	. . . . . . 7 ((a ^ b)_|_ ^ (a v b)_|_) =< (a v b)_|_
10	9	df-le2 123	. . . . . 6 (((a ^ b)_|_ ^ (a v b)_|_) v (a v b)_|_) = (a v b)_|_
11	10	df-le1 122	. . . . 5 ((a ^ b)_|_ ^ (a v b)_|_) =< (a v b)_|_
12	8, 11	lebi 137	. . . 4 (a v b)_|_ = ((a ^ b)_|_ ^ (a v b)_|_)
13	4, 12	ax-r2 35	. . 3 (a_|_ ^ b_|_) = ((a ^ b)_|_ ^ (a v b)_|_)
14	3, 13	2or 67	. 2 ((a ^ b) v (a_|_ ^ b_|_)) = (((a ^ b) ^ (a v b)) v ((a ^ b)_|_ ^ (a v b)_|_))
15		dfb 86	. 2 (a == b) = ((a ^ b) v (a_|_ ^ b_|_))
16		dfb 86	. 2 ((a ^ b) == (a v b)) = (((a ^ b) ^ (a v b)) v ((a ^ b)_|_ ^ (a v b)_|_))
17	14, 15, 16	3tr1 60	1 (a == b) = ((a ^ b) == (a v b))

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

This theorem is referenced by:  mlaconj4 826

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  df-le1 122  df-le2 123

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