Theorem i2bi 704

Description: Dishkant implication expressed with biconditional.

Assertion

Ref	Expression
i2bi	(a ->2 b) = (b v (a == b))

Proof of Theorem i2bi

Step	Hyp	Ref	Expression
1		leor 151	. . . 4 (a_|_ ^ b_|_) =< ((a ^ b) v (a_|_ ^ b_|_))
2	1	lelor 158	. . 3 (b v (a_|_ ^ b_|_)) =< (b v ((a ^ b) v (a_|_ ^ b_|_)))
3		df-i2 44	. . 3 (a ->2 b) = (b v (a_|_ ^ b_|_))
4		dfb 86	. . . 4 (a == b) = ((a ^ b) v (a_|_ ^ b_|_))
5	4	lor 66	. . 3 (b v (a == b)) = (b v ((a ^ b) v (a_|_ ^ b_|_)))
6	2, 3, 5	le3tr1 132	. 2 (a ->2 b) =< (b v (a == b))
7		leo 150	. . . 4 b =< (b v (a_|_ ^ b_|_))
8	3	ax-r1 34	. . . 4 (b v (a_|_ ^ b_|_)) = (a ->2 b)
9	7, 8	lbtr 131	. . 3 b =< (a ->2 b)
10		u2lembi 703	. . . . 5 ((a ->2 b) ^ (b ->2 a)) = (a == b)
11	10	ax-r1 34	. . . 4 (a == b) = ((a ->2 b) ^ (b ->2 a))
12		lea 152	. . . 4 ((a ->2 b) ^ (b ->2 a)) =< (a ->2 b)
13	11, 12	bltr 130	. . 3 (a == b) =< (a ->2 b)
14	9, 13	lel2or 162	. 2 (b v (a == b)) =< (a ->2 b)
15	6, 14	lebi 137	1 (a ->2 b) = (b v (a == b))

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

This theorem is referenced by:  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

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