Theorem u2lembi 703

Description: Dishkant implication and biconditional.

Assertion

Ref	Expression
u2lembi	((a ->2 b) ^ (b ->2 a)) = (a == b)

Proof of Theorem u2lembi

Step	Hyp	Ref	Expression
1		ancom 68	. . 3 ((b v (a_|_ ^ b_|_)) ^ (a v (a_|_ ^ b_|_))) = ((a v (a_|_ ^ b_|_)) ^ (b v (a_|_ ^ b_|_)))
2		coman1 177	. . . . . 6 (a_|_ ^ b_|_) C a_|_
3	2	comcom7 442	. . . . 5 (a_|_ ^ b_|_) C a
4		coman2 178	. . . . . 6 (a_|_ ^ b_|_) C b_|_
5	4	comcom7 442	. . . . 5 (a_|_ ^ b_|_) C b
6	3, 5	fh3r 457	. . . 4 ((a ^ b) v (a_|_ ^ b_|_)) = ((a v (a_|_ ^ b_|_)) ^ (b v (a_|_ ^ b_|_)))
7	6	ax-r1 34	. . 3 ((a v (a_|_ ^ b_|_)) ^ (b v (a_|_ ^ b_|_))) = ((a ^ b) v (a_|_ ^ b_|_))
8	1, 7	ax-r2 35	. 2 ((b v (a_|_ ^ b_|_)) ^ (a v (a_|_ ^ b_|_))) = ((a ^ b) v (a_|_ ^ b_|_))
9		df-i2 44	. . 3 (a ->2 b) = (b v (a_|_ ^ b_|_))
10		df-i2 44	. . . 4 (b ->2 a) = (a v (b_|_ ^ a_|_))
11		ancom 68	. . . . 5 (b_|_ ^ a_|_) = (a_|_ ^ b_|_)
12	11	lor 66	. . . 4 (a v (b_|_ ^ a_|_)) = (a v (a_|_ ^ b_|_))
13	10, 12	ax-r2 35	. . 3 (b ->2 a) = (a v (a_|_ ^ b_|_))
14	9, 13	2an 72	. 2 ((a ->2 b) ^ (b ->2 a)) = ((b v (a_|_ ^ b_|_)) ^ (a v (a_|_ ^ b_|_)))
15		dfb 86	. 2 (a == b) = ((a ^ b) v (a_|_ ^ b_|_))
16	8, 14, 15	3tr1 60	1 ((a ->2 b) ^ (b ->2 a)) = (a == b)

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

This theorem is referenced by:  i2bi 704  mloa 998

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

metamath.org