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Theorem u1lembi 720
Description: Sasaki implication and biconditional.
Assertion
Ref Expression
u1lembi ((a ->1 b) ^ (b ->1 a)) = (a == b)
Proof of Theorem u1lembi
StepHypRef Expression
1 ax-a2 31 . . . 4 (a' v (a ^ b)) = ((a ^ b) v a')
2 ax-a2 31 . . . 4 (b' v (a ^ b)) = ((a ^ b) v b')
31, 22an 79 . . 3 ((a' v (a ^ b)) ^ (b' v (a ^ b))) = (((a ^ b) v a') ^ ((a ^ b) v b'))
4 coman1 185 . . . . . 6 (a ^ b) C a
54comcom2 183 . . . . 5 (a ^ b) C a'
6 coman2 186 . . . . . 6 (a ^ b) C b
76comcom2 183 . . . . 5 (a ^ b) C b'
85, 7fh3 471 . . . 4 ((a ^ b) v (a' ^ b')) = (((a ^ b) v a') ^ ((a ^ b) v b'))
98ax-r1 35 . . 3 (((a ^ b) v a') ^ ((a ^ b) v b')) = ((a ^ b) v (a' ^ b'))
103, 9ax-r2 36 . 2 ((a' v (a ^ b)) ^ (b' v (a ^ b))) = ((a ^ b) v (a' ^ b'))
11 df-i1 44 . . 3 (a ->1 b) = (a' v (a ^ b))
12 df-i1 44 . . . 4 (b ->1 a) = (b' v (b ^ a))
13 ancom 74 . . . . 5 (b ^ a) = (a ^ b)
1413lor 70 . . . 4 (b' v (b ^ a)) = (b' v (a ^ b))
1512, 14ax-r2 36 . . 3 (b ->1 a) = (b' v (a ^ b))
1611, 152an 79 . 2 ((a ->1 b) ^ (b ->1 a)) = ((a' v (a ^ b)) ^ (b' v (a ^ b)))
17 dfb 94 . 2 (a == b) = ((a ^ b) v (a' ^ b'))
1810, 16, 173tr1 63 1 ((a ->1 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
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-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  mlaoml  833  comanblem1  870
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