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Theorem u2lembi 721
Description: Dishkant implication and biconditional.
Assertion
Ref Expression
u2lembi ((a ->2 b) ^ (b ->2 a)) = (a == b)
Proof of Theorem u2lembi
StepHypRef Expression
1 ancom 74 . . 3 ((b v (a' ^ b')) ^ (a v (a' ^ b'))) = ((a v (a' ^ b')) ^ (b v (a' ^ b')))
2 coman1 185 . . . . . 6 (a' ^ b') C a'
32comcom7 460 . . . . 5 (a' ^ b') C a
4 coman2 186 . . . . . 6 (a' ^ b') C b'
54comcom7 460 . . . . 5 (a' ^ b') C b
63, 5fh3r 475 . . . 4 ((a ^ b) v (a' ^ b')) = ((a v (a' ^ b')) ^ (b v (a' ^ b')))
76ax-r1 35 . . 3 ((a v (a' ^ b')) ^ (b v (a' ^ b'))) = ((a ^ b) v (a' ^ b'))
81, 7ax-r2 36 . 2 ((b v (a' ^ b')) ^ (a v (a' ^ b'))) = ((a ^ b) v (a' ^ b'))
9 df-i2 45 . . 3 (a ->2 b) = (b v (a' ^ b'))
10 df-i2 45 . . . 4 (b ->2 a) = (a v (b' ^ a'))
11 ancom 74 . . . . 5 (b' ^ a') = (a' ^ b')
1211lor 70 . . . 4 (a v (b' ^ a')) = (a v (a' ^ b'))
1310, 12ax-r2 36 . . 3 (b ->2 a) = (a v (a' ^ b'))
149, 132an 79 . 2 ((a ->2 b) ^ (b ->2 a)) = ((b v (a' ^ b')) ^ (a v (a' ^ b')))
15 dfb 94 . 2 (a == b) = ((a ^ b) v (a' ^ b'))
168, 14, 153tr1 63 1 ((a ->2 b) ^ (b ->2 a)) = (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:  i2bi  722  mloa  1018
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