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Theorem u4lem4 759
Description: Lemma for unified implication study.
Assertion
Ref Expression
u4lem4 (a ->4 (a ->4 (b ->4 a))) = (a ->4 (b ->4 a))
Proof of Theorem u4lem4
StepHypRef Expression
1 df-i4 47 . 2 (a ->4 (a ->4 (b ->4 a))) = (((a ^ (a ->4 (b ->4 a))) v (a' ^ (a ->4 (b ->4 a)))) v ((a' v (a ->4 (b ->4 a))) ^ (a ->4 (b ->4 a))'))
2 u4lem3 752 . . . . . . . . 9 (a ->4 (b ->4 a)) = (a' v ((a ^ b) v (a ^ b')))
3 comid 187 . . . . . . . . . . . 12 a C a
43comcom2 183 . . . . . . . . . . 11 a C a'
5 comanr1 464 . . . . . . . . . . . 12 a C (a ^ b)
6 comanr1 464 . . . . . . . . . . . 12 a C (a ^ b')
75, 6com2or 483 . . . . . . . . . . 11 a C ((a ^ b) v (a ^ b'))
84, 7com2or 483 . . . . . . . . . 10 a C (a' v ((a ^ b) v (a ^ b')))
98comcom 453 . . . . . . . . 9 (a' v ((a ^ b) v (a ^ b'))) C a
102, 9bctr 181 . . . . . . . 8 (a ->4 (b ->4 a)) C a
1110comcom 453 . . . . . . 7 a C (a ->4 (b ->4 a))
1211, 4fh2r 474 . . . . . 6 ((a v a') ^ (a ->4 (b ->4 a))) = ((a ^ (a ->4 (b ->4 a))) v (a' ^ (a ->4 (b ->4 a))))
1312ax-r1 35 . . . . 5 ((a ^ (a ->4 (b ->4 a))) v (a' ^ (a ->4 (b ->4 a)))) = ((a v a') ^ (a ->4 (b ->4 a)))
14 ancom 74 . . . . . 6 ((a v a') ^ (a ->4 (b ->4 a))) = ((a ->4 (b ->4 a)) ^ (a v a'))
15 df-t 41 . . . . . . . . 9 1 = (a v a')
1615ax-r1 35 . . . . . . . 8 (a v a') = 1
1716lan 77 . . . . . . 7 ((a ->4 (b ->4 a)) ^ (a v a')) = ((a ->4 (b ->4 a)) ^ 1)
18 an1 106 . . . . . . 7 ((a ->4 (b ->4 a)) ^ 1) = (a ->4 (b ->4 a))
1917, 18ax-r2 36 . . . . . 6 ((a ->4 (b ->4 a)) ^ (a v a')) = (a ->4 (b ->4 a))
2014, 19ax-r2 36 . . . . 5 ((a v a') ^ (a ->4 (b ->4 a))) = (a ->4 (b ->4 a))
2113, 20ax-r2 36 . . . 4 ((a ^ (a ->4 (b ->4 a))) v (a' ^ (a ->4 (b ->4 a)))) = (a ->4 (b ->4 a))
2210comcom4 455 . . . . . 6 (a ->4 (b ->4 a))' C a'
23 comid 187 . . . . . . 7 (a ->4 (b ->4 a)) C (a ->4 (b ->4 a))
2423comcom3 454 . . . . . 6 (a ->4 (b ->4 a))' C (a ->4 (b ->4 a))
2522, 24fh1r 473 . . . . 5 ((a' v (a ->4 (b ->4 a))) ^ (a ->4 (b ->4 a))') = ((a' ^ (a ->4 (b ->4 a))') v ((a ->4 (b ->4 a)) ^ (a ->4 (b ->4 a))'))
26 dff 101 . . . . . . . 8 0 = ((a ->4 (b ->4 a)) ^ (a ->4 (b ->4 a))')
2726ax-r1 35 . . . . . . 7 ((a ->4 (b ->4 a)) ^ (a ->4 (b ->4 a))') = 0
2827lor 70 . . . . . 6 ((a' ^ (a ->4 (b ->4 a))') v ((a ->4 (b ->4 a)) ^ (a ->4 (b ->4 a))')) = ((a' ^ (a ->4 (b ->4 a))') v 0)
29 or0 102 . . . . . 6 ((a' ^ (a ->4 (b ->4 a))') v 0) = (a' ^ (a ->4 (b ->4 a))')
3028, 29ax-r2 36 . . . . 5 ((a' ^ (a ->4 (b ->4 a))') v ((a ->4 (b ->4 a)) ^ (a ->4 (b ->4 a))')) = (a' ^ (a ->4 (b ->4 a))')
3125, 30ax-r2 36 . . . 4 ((a' v (a ->4 (b ->4 a))) ^ (a ->4 (b ->4 a))') = (a' ^ (a ->4 (b ->4 a))')
3221, 312or 72 . . 3 (((a ^ (a ->4 (b ->4 a))) v (a' ^ (a ->4 (b ->4 a)))) v ((a' v (a ->4 (b ->4 a))) ^ (a ->4 (b ->4 a))')) = ((a ->4 (b ->4 a)) v (a' ^ (a ->4 (b ->4 a))'))
3310comcom2 183 . . . . . 6 (a ->4 (b ->4 a)) C a'
3423comcom2 183 . . . . . 6 (a ->4 (b ->4 a)) C (a ->4 (b ->4 a))'
3533, 34fh3 471 . . . . 5 ((a ->4 (b ->4 a)) v (a' ^ (a ->4 (b ->4 a))')) = (((a ->4 (b ->4 a)) v a') ^ ((a ->4 (b ->4 a)) v (a ->4 (b ->4 a))'))
36 df-t 41 . . . . . . . 8 1 = ((a ->4 (b ->4 a)) v (a ->4 (b ->4 a))')
3736ax-r1 35 . . . . . . 7 ((a ->4 (b ->4 a)) v (a ->4 (b ->4 a))') = 1
3837lan 77 . . . . . 6 (((a ->4 (b ->4 a)) v a') ^ ((a ->4 (b ->4 a)) v (a ->4 (b ->4 a))')) = (((a ->4 (b ->4 a)) v a') ^ 1)
39 an1 106 . . . . . 6 (((a ->4 (b ->4 a)) v a') ^ 1) = ((a ->4 (b ->4 a)) v a')
4038, 39ax-r2 36 . . . . 5 (((a ->4 (b ->4 a)) v a') ^ ((a ->4 (b ->4 a)) v (a ->4 (b ->4 a))')) = ((a ->4 (b ->4 a)) v a')
4135, 40ax-r2 36 . . . 4 ((a ->4 (b ->4 a)) v (a' ^ (a ->4 (b ->4 a))')) = ((a ->4 (b ->4 a)) v a')
422ax-r5 38 . . . . 5 ((a ->4 (b ->4 a)) v a') = ((a' v ((a ^ b) v (a ^ b'))) v a')
43 or32 82 . . . . . 6 ((a' v ((a ^ b) v (a ^ b'))) v a') = ((a' v a') v ((a ^ b) v (a ^ b')))
44 oridm 110 . . . . . . . 8 (a' v a') = a'
4544ax-r5 38 . . . . . . 7 ((a' v a') v ((a ^ b) v (a ^ b'))) = (a' v ((a ^ b) v (a ^ b')))
462ax-r1 35 . . . . . . 7 (a' v ((a ^ b) v (a ^ b'))) = (a ->4 (b ->4 a))
4745, 46ax-r2 36 . . . . . 6 ((a' v a') v ((a ^ b) v (a ^ b'))) = (a ->4 (b ->4 a))
4843, 47ax-r2 36 . . . . 5 ((a' v ((a ^ b) v (a ^ b'))) v a') = (a ->4 (b ->4 a))
4942, 48ax-r2 36 . . . 4 ((a ->4 (b ->4 a)) v a') = (a ->4 (b ->4 a))
5041, 49ax-r2 36 . . 3 ((a ->4 (b ->4 a)) v (a' ^ (a ->4 (b ->4 a))')) = (a ->4 (b ->4 a))
5132, 50ax-r2 36 . 2 (((a ^ (a ->4 (b ->4 a))) v (a' ^ (a ->4 (b ->4 a)))) v ((a' v (a ->4 (b ->4 a))) ^ (a ->4 (b ->4 a))')) = (a ->4 (b ->4 a))
521, 51ax-r2 36 1 (a ->4 (a ->4 (b ->4 a))) = (a ->4 (b ->4 a))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8  0wf 9   ->4 wi4 15
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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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