Theorem u12lembi 708

Description: Sasaki/Dishkant implication and biconditional.

Assertion

Ref	Expression
u12lembi	((a ->1 b) ^ (b ->2 a)) = (a == b)

Proof of Theorem u12lembi

Step	Hyp	Ref	Expression
1		u1lemc1 662	. . . . 5 a C (a ->1 b)
2	1	comcom 435	. . . 4 (a ->1 b) C a
3		lear 153	. . . . . . 7 (b_|_ ^ a_|_) =< a_|_
4		leo 150	. . . . . . . 8 a_|_ =< (a_|_ v (a ^ b))
5		df-i1 43	. . . . . . . . 9 (a ->1 b) = (a_|_ v (a ^ b))
6	5	ax-r1 34	. . . . . . . 8 (a_|_ v (a ^ b)) = (a ->1 b)
7	4, 6	lbtr 131	. . . . . . 7 a_|_ =< (a ->1 b)
8	3, 7	letr 129	. . . . . 6 (b_|_ ^ a_|_) =< (a ->1 b)
9	8	lecom 172	. . . . 5 (b_|_ ^ a_|_) C (a ->1 b)
10	9	comcom 435	. . . 4 (a ->1 b) C (b_|_ ^ a_|_)
11	2, 10	fh1 451	. . 3 ((a ->1 b) ^ (a v (b_|_ ^ a_|_))) = (((a ->1 b) ^ a) v ((a ->1 b) ^ (b_|_ ^ a_|_)))
12		u1lemaa 582	. . . 4 ((a ->1 b) ^ a) = (a ^ b)
13		an12 74	. . . . 5 ((a ->1 b) ^ (b_|_ ^ a_|_)) = (b_|_ ^ ((a ->1 b) ^ a_|_))
14		u1lemana 587	. . . . . 6 ((a ->1 b) ^ a_|_) = a_|_
15	14	lan 70	. . . . 5 (b_|_ ^ ((a ->1 b) ^ a_|_)) = (b_|_ ^ a_|_)
16		ancom 68	. . . . 5 (b_|_ ^ a_|_) = (a_|_ ^ b_|_)
17	13, 15, 16	3tr 62	. . . 4 ((a ->1 b) ^ (b_|_ ^ a_|_)) = (a_|_ ^ b_|_)
18	12, 17	2or 67	. . 3 (((a ->1 b) ^ a) v ((a ->1 b) ^ (b_|_ ^ a_|_))) = ((a ^ b) v (a_|_ ^ b_|_))
19	11, 18	ax-r2 35	. 2 ((a ->1 b) ^ (a v (b_|_ ^ a_|_))) = ((a ^ b) v (a_|_ ^ b_|_))
20		df-i2 44	. . 3 (b ->2 a) = (a v (b_|_ ^ a_|_))
21	20	lan 70	. 2 ((a ->1 b) ^ (b ->2 a)) = ((a ->1 b) ^ (a v (b_|_ ^ a_|_)))
22		dfb 86	. 2 (a == b) = ((a ^ b) v (a_|_ ^ b_|_))
23	19, 21, 22	3tr1 60	1 ((a ->1 b) ^ (b ->2 a)) = (a == b)

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7   ->1 wi1 13   ->2 wi2 14

This theorem is referenced by:  bi3 821  bi4 822

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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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