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Theorem u3lemc4 703
Description: Lemma for Kalmbach implication study.
Hypothesis
Ref Expression
ulemc3.1 a C b
Assertion
Ref Expression
u3lemc4 (a ->3 b) = (a' v b)
Proof of Theorem u3lemc4
StepHypRef Expression
1 df-i3 46 . 2 (a ->3 b) = (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b)))
2 ulemc3.1 . . . . . . . 8 a C b
32comcom3 454 . . . . . . 7 a' C b
42comcom4 455 . . . . . . 7 a' C b'
53, 4fh1 469 . . . . . 6 (a' ^ (b v b')) = ((a' ^ b) v (a' ^ b'))
65ax-r1 35 . . . . 5 ((a' ^ b) v (a' ^ b')) = (a' ^ (b v b'))
7 df-t 41 . . . . . . . 8 1 = (b v b')
87ax-r1 35 . . . . . . 7 (b v b') = 1
98lan 77 . . . . . 6 (a' ^ (b v b')) = (a' ^ 1)
10 an1 106 . . . . . 6 (a' ^ 1) = a'
119, 10ax-r2 36 . . . . 5 (a' ^ (b v b')) = a'
126, 11ax-r2 36 . . . 4 ((a' ^ b) v (a' ^ b')) = a'
13 comid 187 . . . . . . 7 a C a
1413comcom2 183 . . . . . 6 a C a'
1514, 2fh1 469 . . . . 5 (a ^ (a' v b)) = ((a ^ a') v (a ^ b))
16 ax-a2 31 . . . . . 6 ((a ^ a') v (a ^ b)) = ((a ^ b) v (a ^ a'))
17 dff 101 . . . . . . . . 9 0 = (a ^ a')
1817ax-r1 35 . . . . . . . 8 (a ^ a') = 0
1918lor 70 . . . . . . 7 ((a ^ b) v (a ^ a')) = ((a ^ b) v 0)
20 or0 102 . . . . . . 7 ((a ^ b) v 0) = (a ^ b)
2119, 20ax-r2 36 . . . . . 6 ((a ^ b) v (a ^ a')) = (a ^ b)
2216, 21ax-r2 36 . . . . 5 ((a ^ a') v (a ^ b)) = (a ^ b)
2315, 22ax-r2 36 . . . 4 (a ^ (a' v b)) = (a ^ b)
2412, 232or 72 . . 3 (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) = (a' v (a ^ b))
2514, 2fh4 472 . . . 4 (a' v (a ^ b)) = ((a' v a) ^ (a' v b))
26 ancom 74 . . . . 5 ((a' v a) ^ (a' v b)) = ((a' v b) ^ (a' v a))
27 ax-a2 31 . . . . . . . 8 (a' v a) = (a v a')
28 df-t 41 . . . . . . . . 9 1 = (a v a')
2928ax-r1 35 . . . . . . . 8 (a v a') = 1
3027, 29ax-r2 36 . . . . . . 7 (a' v a) = 1
3130lan 77 . . . . . 6 ((a' v b) ^ (a' v a)) = ((a' v b) ^ 1)
32 an1 106 . . . . . 6 ((a' v b) ^ 1) = (a' v b)
3331, 32ax-r2 36 . . . . 5 ((a' v b) ^ (a' v a)) = (a' v b)
3426, 33ax-r2 36 . . . 4 ((a' v a) ^ (a' v b)) = (a' v b)
3525, 34ax-r2 36 . . 3 (a' v (a ^ b)) = (a' v b)
3624, 35ax-r2 36 . 2 (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) = (a' v b)
371, 36ax-r2 36 1 (a ->3 b) = (a' v b)
Colors of variables: term
Syntax hints:   = wb 1   C wc 3  'wn 4   v wo 6   ^ wa 7  1wt 8  0wf 9   ->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:  u3lemle1  712  u3lem1  736  u3lem2  746  u3lem5  763  u3lem6  767  u3lem7  774  u3lem8  783  u3lem9  784
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