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Theorem u12lembi 726
Description: Sasaki/Dishkant implication and biconditional. Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 1, and j is set to 2.
Assertion
Ref Expression
u12lembi ((a ->1 b) ^ (b ->2 a)) = (a == b)
Proof of Theorem u12lembi
StepHypRef Expression
1 u1lemc1 680 . . . . 5 a C (a ->1 b)
21comcom 453 . . . 4 (a ->1 b) C a
3 lear 161 . . . . . . 7 (b' ^ a') =< a'
4 leo 158 . . . . . . . 8 a' =< (a' v (a ^ b))
5 df-i1 44 . . . . . . . . 9 (a ->1 b) = (a' v (a ^ b))
65ax-r1 35 . . . . . . . 8 (a' v (a ^ b)) = (a ->1 b)
74, 6lbtr 139 . . . . . . 7 a' =< (a ->1 b)
83, 7letr 137 . . . . . 6 (b' ^ a') =< (a ->1 b)
98lecom 180 . . . . 5 (b' ^ a') C (a ->1 b)
109comcom 453 . . . 4 (a ->1 b) C (b' ^ a')
112, 10fh1 469 . . 3 ((a ->1 b) ^ (a v (b' ^ a'))) = (((a ->1 b) ^ a) v ((a ->1 b) ^ (b' ^ a')))
12 u1lemaa 600 . . . 4 ((a ->1 b) ^ a) = (a ^ b)
13 an12 81 . . . . 5 ((a ->1 b) ^ (b' ^ a')) = (b' ^ ((a ->1 b) ^ a'))
14 u1lemana 605 . . . . . 6 ((a ->1 b) ^ a') = a'
1514lan 77 . . . . 5 (b' ^ ((a ->1 b) ^ a')) = (b' ^ a')
16 ancom 74 . . . . 5 (b' ^ a') = (a' ^ b')
1713, 15, 163tr 65 . . . 4 ((a ->1 b) ^ (b' ^ a')) = (a' ^ b')
1812, 172or 72 . . 3 (((a ->1 b) ^ a) v ((a ->1 b) ^ (b' ^ a'))) = ((a ^ b) v (a' ^ b'))
1911, 18ax-r2 36 . 2 ((a ->1 b) ^ (a v (b' ^ a'))) = ((a ^ b) v (a' ^ b'))
20 df-i2 45 . . 3 (b ->2 a) = (a v (b' ^ a'))
2120lan 77 . 2 ((a ->1 b) ^ (b ->2 a)) = ((a ->1 b) ^ (a v (b' ^ a')))
22 dfb 94 . 2 (a == b) = ((a ^ b) v (a' ^ b'))
2319, 21, 223tr1 63 1 ((a ->1 b) ^ (b ->2 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:  bi3  839  bi4  840
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