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Theorem u21lembi 727
Description: Dishkant/Sasaki implication and biconditional.
Assertion
Ref Expression
u21lembi ((a ->2 b) ^ (b ->1 a)) = (a == b)
Proof of Theorem u21lembi
StepHypRef Expression
1 u2lemc1 681 . . . . 5 b C (a ->2 b)
21comcom3 454 . . . 4 b' C (a ->2 b)
3 comanr1 464 . . . . 5 b C (b ^ a)
43comcom3 454 . . . 4 b' C (b ^ a)
52, 4fh2 470 . . 3 ((a ->2 b) ^ (b' v (b ^ a))) = (((a ->2 b) ^ b') v ((a ->2 b) ^ (b ^ a)))
6 u2lemanb 616 . . . 4 ((a ->2 b) ^ b') = (a' ^ b')
7 u2lemab 611 . . . . . 6 ((a ->2 b) ^ b) = b
87ran 78 . . . . 5 (((a ->2 b) ^ b) ^ a) = (b ^ a)
9 anass 76 . . . . 5 (((a ->2 b) ^ b) ^ a) = ((a ->2 b) ^ (b ^ a))
10 ancom 74 . . . . 5 (b ^ a) = (a ^ b)
118, 9, 103tr2 64 . . . 4 ((a ->2 b) ^ (b ^ a)) = (a ^ b)
126, 112or 72 . . 3 (((a ->2 b) ^ b') v ((a ->2 b) ^ (b ^ a))) = ((a' ^ b') v (a ^ b))
13 ax-a2 31 . . 3 ((a' ^ b') v (a ^ b)) = ((a ^ b) v (a' ^ b'))
145, 12, 133tr 65 . 2 ((a ->2 b) ^ (b' v (b ^ a))) = ((a ^ b) v (a' ^ b'))
15 df-i1 44 . . 3 (b ->1 a) = (b' v (b ^ a))
1615lan 77 . 2 ((a ->2 b) ^ (b ->1 a)) = ((a ->2 b) ^ (b' v (b ^ a)))
17 dfb 94 . 2 (a == b) = ((a ^ b) v (a' ^ b'))
1814, 16, 173tr1 63 1 ((a ->2 b) ^ (b ->1 a)) = (a == b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7   ->1 wi1 12   ->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-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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