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Theorem lem3.3.7i3e1 1066
Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 3, and this is the first part of the equation. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem3.3.7i3e1 (a ->3 (a ^ b)) = (a ==3 (a ^ b))
Proof of Theorem lem3.3.7i3e1
StepHypRef Expression
1 anass 76 . . . . . 6 ((a' ^ a) ^ b) = (a' ^ (a ^ b))
21ax-r1 35 . . . . 5 (a' ^ (a ^ b)) = ((a' ^ a) ^ b)
32ax-r5 38 . . . 4 ((a' ^ (a ^ b)) v (a' ^ (a ^ b)')) = (((a' ^ a) ^ b) v (a' ^ (a ^ b)'))
43ax-r5 38 . . 3 (((a' ^ (a ^ b)) v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b)))) = ((((a' ^ a) ^ b) v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b))))
5 ancom 74 . . . . . 6 (a' ^ a) = (a ^ a')
65ran 78 . . . . 5 ((a' ^ a) ^ b) = ((a ^ a') ^ b)
76ax-r5 38 . . . 4 (((a' ^ a) ^ b) v (a' ^ (a ^ b)')) = (((a ^ a') ^ b) v (a' ^ (a ^ b)'))
87ax-r5 38 . . 3 ((((a' ^ a) ^ b) v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b)))) = ((((a ^ a') ^ b) v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b))))
9 dff 101 . . . . . . . 8 0 = (a ^ a')
109ax-r1 35 . . . . . . 7 (a ^ a') = 0
1110ran 78 . . . . . 6 ((a ^ a') ^ b) = (0 ^ b)
1211ax-r5 38 . . . . 5 (((a ^ a') ^ b) v (a' ^ (a ^ b)')) = ((0 ^ b) v (a' ^ (a ^ b)'))
1312ax-r5 38 . . . 4 ((((a ^ a') ^ b) v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b)))) = (((0 ^ b) v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b))))
14 an0r 109 . . . . . 6 (0 ^ b) = 0
1514ax-r5 38 . . . . 5 ((0 ^ b) v (a' ^ (a ^ b)')) = (0 v (a' ^ (a ^ b)'))
1615ax-r5 38 . . . 4 (((0 ^ b) v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b)))) = ((0 v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b))))
17 or0r 103 . . . . . 6 (0 v (a' ^ (a ^ b)')) = (a' ^ (a ^ b)')
1817ax-r5 38 . . . . 5 ((0 v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b)))) = ((a' ^ (a ^ b)') v (a ^ (a' v (a ^ b))))
19 anor3 90 . . . . . . 7 (a' ^ (a ^ b)') = (a v (a ^ b))'
2019ax-r5 38 . . . . . 6 ((a' ^ (a ^ b)') v (a ^ (a' v (a ^ b)))) = ((a v (a ^ b))' v (a ^ (a' v (a ^ b))))
21 orabs 120 . . . . . . . 8 (a v (a ^ b)) = a
2221ax-r4 37 . . . . . . 7 (a v (a ^ b))' = a'
2322ax-r5 38 . . . . . 6 ((a v (a ^ b))' v (a ^ (a' v (a ^ b)))) = (a' v (a ^ (a' v (a ^ b))))
24 womaa 222 . . . . . . . 8 (a' v (a ^ (a' v (a ^ b)))) = (a' v (a ^ b))
25 an1 106 . . . . . . . . 9 ((a' v (a ^ b)) ^ 1) = (a' v (a ^ b))
2625ax-r1 35 . . . . . . . 8 (a' v (a ^ b)) = ((a' v (a ^ b)) ^ 1)
27 df-t 41 . . . . . . . . 9 1 = (a v a')
2827lan 77 . . . . . . . 8 ((a' v (a ^ b)) ^ 1) = ((a' v (a ^ b)) ^ (a v a'))
2924, 26, 283tr 65 . . . . . . 7 (a' v (a ^ (a' v (a ^ b)))) = ((a' v (a ^ b)) ^ (a v a'))
3021ax-r1 35 . . . . . . . . . 10 a = (a v (a ^ b))
3130ax-r4 37 . . . . . . . . 9 a' = (a v (a ^ b))'
3231lor 70 . . . . . . . 8 (a v a') = (a v (a v (a ^ b))')
3332lan 77 . . . . . . 7 ((a' v (a ^ b)) ^ (a v a')) = ((a' v (a ^ b)) ^ (a v (a v (a ^ b))'))
3419ax-r1 35 . . . . . . . . 9 (a v (a ^ b))' = (a' ^ (a ^ b)')
3534lor 70 . . . . . . . 8 (a v (a v (a ^ b))') = (a v (a' ^ (a ^ b)'))
3635lan 77 . . . . . . 7 ((a' v (a ^ b)) ^ (a v (a v (a ^ b))')) = ((a' v (a ^ b)) ^ (a v (a' ^ (a ^ b)')))
3729, 33, 363tr 65 . . . . . 6 (a' v (a ^ (a' v (a ^ b)))) = ((a' v (a ^ b)) ^ (a v (a' ^ (a ^ b)')))
3820, 23, 373tr 65 . . . . 5 ((a' ^ (a ^ b)') v (a ^ (a' v (a ^ b)))) = ((a' v (a ^ b)) ^ (a v (a' ^ (a ^ b)')))
3918, 38ax-r2 36 . . . 4 ((0 v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b)))) = ((a' v (a ^ b)) ^ (a v (a' ^ (a ^ b)')))
4013, 16, 393tr 65 . . 3 ((((a ^ a') ^ b) v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b)))) = ((a' v (a ^ b)) ^ (a v (a' ^ (a ^ b)')))
414, 8, 403tr 65 . 2 (((a' ^ (a ^ b)) v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b)))) = ((a' v (a ^ b)) ^ (a v (a' ^ (a ^ b)')))
42 df-i3 46 . 2 (a ->3 (a ^ b)) = (((a' ^ (a ^ b)) v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b))))
43 df-id3 52 . 2 (a ==3 (a ^ b)) = ((a' v (a ^ b)) ^ (a v (a' ^ (a ^ b)')))
4441, 42, 433tr1 63 1 (a ->3 (a ^ b)) = (a ==3 (a ^ b))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8  0wf 9   ->3 wi3 14   ==3 wid3 20
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-id3 52  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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