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Theorem i2bi 722
Description: Dishkant implication expressed with biconditional.
Assertion
Ref Expression
i2bi (a ->2 b) = (b v (a == b))
Proof of Theorem i2bi
StepHypRef Expression
1 leor 159 . . . 4 (a' ^ b') =< ((a ^ b) v (a' ^ b'))
21lelor 166 . . 3 (b v (a' ^ b')) =< (b v ((a ^ b) v (a' ^ b')))
3 df-i2 45 . . 3 (a ->2 b) = (b v (a' ^ b'))
4 dfb 94 . . . 4 (a == b) = ((a ^ b) v (a' ^ b'))
54lor 70 . . 3 (b v (a == b)) = (b v ((a ^ b) v (a' ^ b')))
62, 3, 5le3tr1 140 . 2 (a ->2 b) =< (b v (a == b))
7 leo 158 . . . 4 b =< (b v (a' ^ b'))
83ax-r1 35 . . . 4 (b v (a' ^ b')) = (a ->2 b)
97, 8lbtr 139 . . 3 b =< (a ->2 b)
10 u2lembi 721 . . . . 5 ((a ->2 b) ^ (b ->2 a)) = (a == b)
1110ax-r1 35 . . . 4 (a == b) = ((a ->2 b) ^ (b ->2 a))
12 lea 160 . . . 4 ((a ->2 b) ^ (b ->2 a)) =< (a ->2 b)
1311, 12bltr 138 . . 3 (a == b) =< (a ->2 b)
149, 13lel2or 170 . 2 (b v (a == b)) =< (a ->2 b)
156, 14lebi 145 1 (a ->2 b) = (b v (a == b))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7   ->2 wi2 13
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  mloa  1018
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