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Theorem u3lem10 785
Description: Lemma for unified implication study.
Assertion
Ref Expression
u3lem10 (a ->3 (a' ^ (a v b))) = a'
Proof of Theorem u3lem10
StepHypRef Expression
1 df-i3 46 . 2 (a ->3 (a' ^ (a v b))) = (((a' ^ (a' ^ (a v b))) v (a' ^ (a' ^ (a v b))')) v (a ^ (a' v (a' ^ (a v b)))))
2 anass 76 . . . . . . . 8 ((a' ^ a') ^ (a v b)) = (a' ^ (a' ^ (a v b)))
32ax-r1 35 . . . . . . 7 (a' ^ (a' ^ (a v b))) = ((a' ^ a') ^ (a v b))
4 anidm 111 . . . . . . . 8 (a' ^ a') = a'
54ran 78 . . . . . . 7 ((a' ^ a') ^ (a v b)) = (a' ^ (a v b))
63, 5ax-r2 36 . . . . . 6 (a' ^ (a' ^ (a v b))) = (a' ^ (a v b))
7 anor3 90 . . . . . . . . . . 11 (a' ^ b') = (a v b)'
87lor 70 . . . . . . . . . 10 (a v (a' ^ b')) = (a v (a v b)')
9 oran1 91 . . . . . . . . . 10 (a v (a v b)') = (a' ^ (a v b))'
108, 9ax-r2 36 . . . . . . . . 9 (a v (a' ^ b')) = (a' ^ (a v b))'
1110ax-r1 35 . . . . . . . 8 (a' ^ (a v b))' = (a v (a' ^ b'))
1211lan 77 . . . . . . 7 (a' ^ (a' ^ (a v b))') = (a' ^ (a v (a' ^ b')))
13 omlan 448 . . . . . . 7 (a' ^ (a v (a' ^ b'))) = (a' ^ b')
1412, 13ax-r2 36 . . . . . 6 (a' ^ (a' ^ (a v b))') = (a' ^ b')
156, 142or 72 . . . . 5 ((a' ^ (a' ^ (a v b))) v (a' ^ (a' ^ (a v b))')) = ((a' ^ (a v b)) v (a' ^ b'))
16 comanr1 464 . . . . . . 7 a' C (a' ^ b')
17 comorr 184 . . . . . . . 8 a C (a v b)
1817comcom3 454 . . . . . . 7 a' C (a v b)
1916, 18fh4r 476 . . . . . 6 ((a' ^ (a v b)) v (a' ^ b')) = ((a' v (a' ^ b')) ^ ((a v b) v (a' ^ b')))
20 orabs 120 . . . . . . . 8 (a' v (a' ^ b')) = a'
217lor 70 . . . . . . . . 9 ((a v b) v (a' ^ b')) = ((a v b) v (a v b)')
22 df-t 41 . . . . . . . . . 10 1 = ((a v b) v (a v b)')
2322ax-r1 35 . . . . . . . . 9 ((a v b) v (a v b)') = 1
2421, 23ax-r2 36 . . . . . . . 8 ((a v b) v (a' ^ b')) = 1
2520, 242an 79 . . . . . . 7 ((a' v (a' ^ b')) ^ ((a v b) v (a' ^ b'))) = (a' ^ 1)
26 an1 106 . . . . . . 7 (a' ^ 1) = a'
2725, 26ax-r2 36 . . . . . 6 ((a' v (a' ^ b')) ^ ((a v b) v (a' ^ b'))) = a'
2819, 27ax-r2 36 . . . . 5 ((a' ^ (a v b)) v (a' ^ b')) = a'
2915, 28ax-r2 36 . . . 4 ((a' ^ (a' ^ (a v b))) v (a' ^ (a' ^ (a v b))')) = a'
30 orabs 120 . . . . . 6 (a' v (a' ^ (a v b))) = a'
3130lan 77 . . . . 5 (a ^ (a' v (a' ^ (a v b)))) = (a ^ a')
32 ancom 74 . . . . 5 (a ^ a') = (a' ^ a)
3331, 32ax-r2 36 . . . 4 (a ^ (a' v (a' ^ (a v b)))) = (a' ^ a)
3429, 332or 72 . . 3 (((a' ^ (a' ^ (a v b))) v (a' ^ (a' ^ (a v b))')) v (a ^ (a' v (a' ^ (a v b))))) = (a' v (a' ^ a))
35 orabs 120 . . 3 (a' v (a' ^ a)) = a'
3634, 35ax-r2 36 . 2 (((a' ^ (a' ^ (a v b))) v (a' ^ (a' ^ (a v b))')) v (a ^ (a' v (a' ^ (a v b))))) = a'
371, 36ax-r2 36 1 (a ->3 (a' ^ (a v b))) = a'
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: (None)
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