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Theorem u1lem4 757
Description: Lemma for unified implication study.
Assertion
Ref Expression
u1lem4 (a ->1 (a ->1 (b ->1 a))) = (a ->1 (b ->1 a))
Proof of Theorem u1lem4
StepHypRef Expression
1 df-i1 44 . 2 (a ->1 (a ->1 (b ->1 a))) = (a' v (a ^ (a ->1 (b ->1 a))))
2 comid 187 . . . . 5 a C a
32comcom2 183 . . . 4 a C a'
4 u1lemc1 680 . . . 4 a C (a ->1 (b ->1 a))
53, 4fh4 472 . . 3 (a' v (a ^ (a ->1 (b ->1 a)))) = ((a' v a) ^ (a' v (a ->1 (b ->1 a))))
6 ax-a2 31 . . . . . 6 (a' v a) = (a v a')
7 df-t 41 . . . . . . 7 1 = (a v a')
87ax-r1 35 . . . . . 6 (a v a') = 1
96, 8ax-r2 36 . . . . 5 (a' v a) = 1
10 ax-a2 31 . . . . . 6 (a' v (a ->1 (b ->1 a))) = ((a ->1 (b ->1 a)) v a')
11 u1lemona 625 . . . . . . 7 ((a ->1 (b ->1 a)) v a') = (a' v (a ^ (b ->1 a)))
12 df-i1 44 . . . . . . . . . . 11 (b ->1 a) = (b' v (b ^ a))
13 ancom 74 . . . . . . . . . . . 12 (b ^ a) = (a ^ b)
1413lor 70 . . . . . . . . . . 11 (b' v (b ^ a)) = (b' v (a ^ b))
1512, 14ax-r2 36 . . . . . . . . . 10 (b ->1 a) = (b' v (a ^ b))
1615lan 77 . . . . . . . . 9 (a ^ (b ->1 a)) = (a ^ (b' v (a ^ b)))
1716lor 70 . . . . . . . 8 (a' v (a ^ (b ->1 a))) = (a' v (a ^ (b' v (a ^ b))))
18 u1lem3 749 . . . . . . . . . 10 (a ->1 (b ->1 a)) = (a' v ((a ^ b) v (a ^ b')))
19 ax-a2 31 . . . . . . . . . . . . . 14 (b' v (a ^ b)) = ((a ^ b) v b')
2019lan 77 . . . . . . . . . . . . 13 (a ^ (b' v (a ^ b))) = (a ^ ((a ^ b) v b'))
21 coman1 185 . . . . . . . . . . . . . . 15 (a ^ b) C a
22 coman2 186 . . . . . . . . . . . . . . . 16 (a ^ b) C b
2322comcom2 183 . . . . . . . . . . . . . . 15 (a ^ b) C b'
2421, 23fh2 470 . . . . . . . . . . . . . 14 (a ^ ((a ^ b) v b')) = ((a ^ (a ^ b)) v (a ^ b'))
25 anass 76 . . . . . . . . . . . . . . . . 17 ((a ^ a) ^ b) = (a ^ (a ^ b))
2625ax-r1 35 . . . . . . . . . . . . . . . 16 (a ^ (a ^ b)) = ((a ^ a) ^ b)
27 anidm 111 . . . . . . . . . . . . . . . . 17 (a ^ a) = a
2827ran 78 . . . . . . . . . . . . . . . 16 ((a ^ a) ^ b) = (a ^ b)
2926, 28ax-r2 36 . . . . . . . . . . . . . . 15 (a ^ (a ^ b)) = (a ^ b)
3029ax-r5 38 . . . . . . . . . . . . . 14 ((a ^ (a ^ b)) v (a ^ b')) = ((a ^ b) v (a ^ b'))
3124, 30ax-r2 36 . . . . . . . . . . . . 13 (a ^ ((a ^ b) v b')) = ((a ^ b) v (a ^ b'))
3220, 31ax-r2 36 . . . . . . . . . . . 12 (a ^ (b' v (a ^ b))) = ((a ^ b) v (a ^ b'))
3332ax-r1 35 . . . . . . . . . . 11 ((a ^ b) v (a ^ b')) = (a ^ (b' v (a ^ b)))
3433lor 70 . . . . . . . . . 10 (a' v ((a ^ b) v (a ^ b'))) = (a' v (a ^ (b' v (a ^ b))))
3518, 34ax-r2 36 . . . . . . . . 9 (a ->1 (b ->1 a)) = (a' v (a ^ (b' v (a ^ b))))
3635ax-r1 35 . . . . . . . 8 (a' v (a ^ (b' v (a ^ b)))) = (a ->1 (b ->1 a))
3717, 36ax-r2 36 . . . . . . 7 (a' v (a ^ (b ->1 a))) = (a ->1 (b ->1 a))
3811, 37ax-r2 36 . . . . . 6 ((a ->1 (b ->1 a)) v a') = (a ->1 (b ->1 a))
3910, 38ax-r2 36 . . . . 5 (a' v (a ->1 (b ->1 a))) = (a ->1 (b ->1 a))
409, 392an 79 . . . 4 ((a' v a) ^ (a' v (a ->1 (b ->1 a)))) = (1 ^ (a ->1 (b ->1 a)))
41 ancom 74 . . . . 5 (1 ^ (a ->1 (b ->1 a))) = ((a ->1 (b ->1 a)) ^ 1)
42 an1 106 . . . . 5 ((a ->1 (b ->1 a)) ^ 1) = (a ->1 (b ->1 a))
4341, 42ax-r2 36 . . . 4 (1 ^ (a ->1 (b ->1 a))) = (a ->1 (b ->1 a))
4440, 43ax-r2 36 . . 3 ((a' v a) ^ (a' v (a ->1 (b ->1 a)))) = (a ->1 (b ->1 a))
455, 44ax-r2 36 . 2 (a' v (a ^ (a ->1 (b ->1 a)))) = (a ->1 (b ->1 a))
461, 45ax-r2 36 1 (a ->1 (a ->1 (b ->1 a))) = (a ->1 (b ->1 a))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->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: (None)
  Copyright terms: Public domain W3C validator