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Theorem i3th1 543
Description: Theorem for Kalmbach implication.
Assertion
Ref Expression
i3th1 (a ->3 (a ->3 (b ->3 a))) = 1
Proof of Theorem i3th1
StepHypRef Expression
1 df2i3 498 . . 3 (b ->3 a) = ((b' ^ a') v ((b' v a) ^ (b v (b' ^ a))))
21lor 70 . 2 (a' v (b ->3 a)) = (a' v ((b' ^ a') v ((b' v a) ^ (b v (b' ^ a)))))
3 lem4 511 . 2 (a ->3 (a ->3 (b ->3 a))) = (a' v (b ->3 a))
4 ax-a3 32 . . . . 5 ((a' v b) v (a ^ b')) = (a' v (b v (a ^ b')))
5 anor1 88 . . . . . 6 (a ^ b') = (a' v b)'
65lor 70 . . . . 5 ((a' v b) v (a ^ b')) = ((a' v b) v (a' v b)')
7 ax-a3 32 . . . . . . . 8 ((a' v (a' ^ b)) v ((b' v a) ^ (b v (b' ^ a)))) = (a' v ((a' ^ b) v ((b' v a) ^ (b v (b' ^ a)))))
8 ax-a2 31 . . . . . . . . . . . . 13 (b' v a) = (a v b')
9 anor2 89 . . . . . . . . . . . . . . 15 (a' ^ b) = (a v b')'
109con2 67 . . . . . . . . . . . . . 14 (a' ^ b)' = (a v b')
1110ax-r1 35 . . . . . . . . . . . . 13 (a v b') = (a' ^ b)'
128, 11ax-r2 36 . . . . . . . . . . . 12 (b' v a) = (a' ^ b)'
13 ancom 74 . . . . . . . . . . . . 13 (b' ^ a) = (a ^ b')
1413lor 70 . . . . . . . . . . . 12 (b v (b' ^ a)) = (b v (a ^ b'))
1512, 142an 79 . . . . . . . . . . 11 ((b' v a) ^ (b v (b' ^ a))) = ((a' ^ b)' ^ (b v (a ^ b')))
1615lor 70 . . . . . . . . . 10 ((a' ^ b) v ((b' v a) ^ (b v (b' ^ a)))) = ((a' ^ b) v ((a' ^ b)' ^ (b v (a ^ b'))))
17 oml5 449 . . . . . . . . . 10 ((a' ^ b) v ((a' ^ b)' ^ (b v (a ^ b')))) = (b v (a ^ b'))
1816, 17ax-r2 36 . . . . . . . . 9 ((a' ^ b) v ((b' v a) ^ (b v (b' ^ a)))) = (b v (a ^ b'))
1918lor 70 . . . . . . . 8 (a' v ((a' ^ b) v ((b' v a) ^ (b v (b' ^ a))))) = (a' v (b v (a ^ b')))
207, 19ax-r2 36 . . . . . . 7 ((a' v (a' ^ b)) v ((b' v a) ^ (b v (b' ^ a)))) = (a' v (b v (a ^ b')))
2120ax-r1 35 . . . . . 6 (a' v (b v (a ^ b'))) = ((a' v (a' ^ b)) v ((b' v a) ^ (b v (b' ^ a))))
22 orabs 120 . . . . . . 7 (a' v (a' ^ b)) = a'
2322ax-r5 38 . . . . . 6 ((a' v (a' ^ b)) v ((b' v a) ^ (b v (b' ^ a)))) = (a' v ((b' v a) ^ (b v (b' ^ a))))
2421, 23ax-r2 36 . . . . 5 (a' v (b v (a ^ b'))) = (a' v ((b' v a) ^ (b v (b' ^ a))))
254, 6, 243tr2 64 . . . 4 ((a' v b) v (a' v b)') = (a' v ((b' v a) ^ (b v (b' ^ a))))
26 df-t 41 . . . 4 1 = ((a' v b) v (a' v b)')
27 ancom 74 . . . . . . 7 (b' ^ a') = (a' ^ b')
2827lor 70 . . . . . 6 (a' v (b' ^ a')) = (a' v (a' ^ b'))
29 orabs 120 . . . . . 6 (a' v (a' ^ b')) = a'
3028, 29ax-r2 36 . . . . 5 (a' v (b' ^ a')) = a'
3130ax-r5 38 . . . 4 ((a' v (b' ^ a')) v ((b' v a) ^ (b v (b' ^ a)))) = (a' v ((b' v a) ^ (b v (b' ^ a))))
3225, 26, 313tr1 63 . . 3 1 = ((a' v (b' ^ a')) v ((b' v a) ^ (b v (b' ^ a))))
33 ax-a3 32 . . 3 ((a' v (b' ^ a')) v ((b' v a) ^ (b v (b' ^ a)))) = (a' v ((b' ^ a') v ((b' v a) ^ (b v (b' ^ a)))))
3432, 33ax-r2 36 . 2 1 = (a' v ((b' ^ a') v ((b' v a) ^ (b v (b' ^ a)))))
352, 3, 343tr1 63 1 (a ->3 (a ->3 (b ->3 a))) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->3 wi3 14
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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  u3lem14aa  792
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