Theorem bi1o1a 780

Description: Equivalence to biconditional.

Assertion

Ref	Expression
bi1o1a	(a == b) = ((a ->1 (a ^ b)) ^ ((a v b) ->1 a))

Proof of Theorem bi1o1a

Step	Hyp	Ref	Expression
1		lea 152	. . . . . . 7 (a_|_ ^ b_|_) =< a_|_
2		leo 150	. . . . . . 7 a_|_ =< (a_|_ v (a ^ b))
3	1, 2	letr 129	. . . . . 6 (a_|_ ^ b_|_) =< (a_|_ v (a ^ b))
4	3	lecom 172	. . . . 5 (a_|_ ^ b_|_) C (a_|_ v (a ^ b))
5	4	comcom 435	. . . 4 (a_|_ v (a ^ b)) C (a_|_ ^ b_|_)
6		comor1 443	. . . . 5 (a_|_ v (a ^ b)) C a_|_
7	6	comcom7 442	. . . 4 (a_|_ v (a ^ b)) C a
8	5, 7	fh1 451	. . 3 ((a_|_ v (a ^ b)) ^ ((a_|_ ^ b_|_) v a)) = (((a_|_ v (a ^ b)) ^ (a_|_ ^ b_|_)) v ((a_|_ v (a ^ b)) ^ a))
9	8	ax-r1 34	. 2 (((a_|_ v (a ^ b)) ^ (a_|_ ^ b_|_)) v ((a_|_ v (a ^ b)) ^ a)) = ((a_|_ v (a ^ b)) ^ ((a_|_ ^ b_|_) v a))
10		dfb 86	. . 3 (a == b) = ((a ^ b) v (a_|_ ^ b_|_))
11		ax-a2 30	. . 3 ((a ^ b) v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v (a ^ b))
12		leid 140	. . . . . 6 (a_|_ ^ b_|_) =< (a_|_ ^ b_|_)
13	3, 12	ler2an 165	. . . . 5 (a_|_ ^ b_|_) =< ((a_|_ v (a ^ b)) ^ (a_|_ ^ b_|_))
14		lear 153	. . . . 5 ((a_|_ v (a ^ b)) ^ (a_|_ ^ b_|_)) =< (a_|_ ^ b_|_)
15	13, 14	lebi 137	. . . 4 (a_|_ ^ b_|_) = ((a_|_ v (a ^ b)) ^ (a_|_ ^ b_|_))
16		dff 93	. . . . . . 7 0 = (a ^ a_|_)
17		ancom 68	. . . . . . 7 (a ^ a_|_) = (a_|_ ^ a)
18	16, 17	ax-r2 35	. . . . . 6 0 = (a_|_ ^ a)
19	18	ax-r5 37	. . . . 5 (0 v ((a ^ b) ^ a)) = ((a_|_ ^ a) v ((a ^ b) ^ a))
20		lea 152	. . . . . . . 8 (a ^ b) =< a
21	20	df2le2 128	. . . . . . 7 ((a ^ b) ^ a) = (a ^ b)
22	21	ax-r1 34	. . . . . 6 (a ^ b) = ((a ^ b) ^ a)
23		or0r 95	. . . . . . 7 (0 v ((a ^ b) ^ a)) = ((a ^ b) ^ a)
24	23	ax-r1 34	. . . . . 6 ((a ^ b) ^ a) = (0 v ((a ^ b) ^ a))
25	22, 24	ax-r2 35	. . . . 5 (a ^ b) = (0 v ((a ^ b) ^ a))
26		comid 179	. . . . . . 7 a C a
27	26	comcom2 175	. . . . . 6 a C a_|_
28		comanr1 446	. . . . . 6 a C (a ^ b)
29	27, 28	fh1r 455	. . . . 5 ((a_|_ v (a ^ b)) ^ a) = ((a_|_ ^ a) v ((a ^ b) ^ a))
30	19, 25, 29	3tr1 60	. . . 4 (a ^ b) = ((a_|_ v (a ^ b)) ^ a)
31	15, 30	2or 67	. . 3 ((a_|_ ^ b_|_) v (a ^ b)) = (((a_|_ v (a ^ b)) ^ (a_|_ ^ b_|_)) v ((a_|_ v (a ^ b)) ^ a))
32	10, 11, 31	3tr 62	. 2 (a == b) = (((a_|_ v (a ^ b)) ^ (a_|_ ^ b_|_)) v ((a_|_ v (a ^ b)) ^ a))
33		df-i1 43	. . . 4 (a ->1 (a ^ b)) = (a_|_ v (a ^ (a ^ b)))
34		lear 153	. . . . . 6 (a ^ (a ^ b)) =< (a ^ b)
35		leid 140	. . . . . . 7 (a ^ b) =< (a ^ b)
36	20, 35	ler2an 165	. . . . . 6 (a ^ b) =< (a ^ (a ^ b))
37	34, 36	lebi 137	. . . . 5 (a ^ (a ^ b)) = (a ^ b)
38	37	lor 66	. . . 4 (a_|_ v (a ^ (a ^ b))) = (a_|_ v (a ^ b))
39	33, 38	ax-r2 35	. . 3 (a ->1 (a ^ b)) = (a_|_ v (a ^ b))
40		df-i1 43	. . . 4 ((a v b) ->1 a) = ((a v b)_|_ v ((a v b) ^ a))
41		anor3 82	. . . . . 6 (a_|_ ^ b_|_) = (a v b)_|_
42	41	ax-r1 34	. . . . 5 (a v b)_|_ = (a_|_ ^ b_|_)
43		lear 153	. . . . . 6 ((a v b) ^ a) =< a
44		leo 150	. . . . . . 7 a =< (a v b)
45		leid 140	. . . . . . 7 a =< a
46	44, 45	ler2an 165	. . . . . 6 a =< ((a v b) ^ a)
47	43, 46	lebi 137	. . . . 5 ((a v b) ^ a) = a
48	42, 47	2or 67	. . . 4 ((a v b)_|_ v ((a v b) ^ a)) = ((a_|_ ^ b_|_) v a)
49	40, 48	ax-r2 35	. . 3 ((a v b) ->1 a) = ((a_|_ ^ b_|_) v a)
50	39, 49	2an 72	. 2 ((a ->1 (a ^ b)) ^ ((a v b) ->1 a)) = ((a_|_ v (a ^ b)) ^ ((a_|_ ^ b_|_) v a))
51	9, 32, 50	3tr1 60	1 (a == b) = ((a ->1 (a ^ b)) ^ ((a v b) ->1 a))

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7  0wf 10   ->1 wi1 13

This theorem is referenced by:  mlaconj 827

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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