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Theorem lei3 246
Description: L.e. to Kalmbach implication.
Hypothesis
Ref Expression
lei3.1 a =< b
Assertion
Ref Expression
lei3 (a ->3 b) = 1
Proof of Theorem lei3
StepHypRef Expression
1 ax-a3 32 . . 3 (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) = ((a' ^ b) v ((a' ^ b') v (a ^ (a' v b))))
2 ax-a2 31 . . . . 5 (b' v a) = (a v b')
3 ancom 74 . . . . . . 7 (a' ^ b') = (b' ^ a')
4 lei3.1 . . . . . . . . 9 a =< b
54lecon 154 . . . . . . . 8 b' =< a'
65df2le2 136 . . . . . . 7 (b' ^ a') = b'
73, 6ax-r2 36 . . . . . 6 (a' ^ b') = b'
84sklem 230 . . . . . . . 8 (a' v b) = 1
98lan 77 . . . . . . 7 (a ^ (a' v b)) = (a ^ 1)
10 an1 106 . . . . . . 7 (a ^ 1) = a
119, 10ax-r2 36 . . . . . 6 (a ^ (a' v b)) = a
127, 112or 72 . . . . 5 ((a' ^ b') v (a ^ (a' v b))) = (b' v a)
13 anor2 89 . . . . . 6 (a' ^ b) = (a v b')'
1413con2 67 . . . . 5 (a' ^ b)' = (a v b')
152, 12, 143tr1 63 . . . 4 ((a' ^ b') v (a ^ (a' v b))) = (a' ^ b)'
1615lor 70 . . 3 ((a' ^ b) v ((a' ^ b') v (a ^ (a' v b)))) = ((a' ^ b) v (a' ^ b)')
171, 16ax-r2 36 . 2 (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) = ((a' ^ b) v (a' ^ b)')
18 df-i3 46 . 2 (a ->3 b) = (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b)))
19 df-t 41 . 2 1 = ((a' ^ b) v (a' ^ b)')
2017, 18, 193tr1 63 1 (a ->3 b) = 1
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  '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
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131
This theorem is referenced by:  bina3  284  bina4  285  bina5  286  bii3  516  binr1  517  binr2  518  binr3  519  i3ri3  538  i3li3  539  i32i3  540  i3th5  547  i3th7  549  i3th8  550
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