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Theorem lem3.3.7i0e1 1057
Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 0, and this is the first part of the equation. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem3.3.7i0e1 (a ->0 (a ^ b)) = (a ==0 (a ^ b))
Proof of Theorem lem3.3.7i0e1
StepHypRef Expression
1 or1 104 . . . . . . 7 (b' v 1) = 1
21ax-r1 35 . . . . . 6 1 = (b' v 1)
32lan 77 . . . . 5 ((a' v (a ^ b)) ^ 1) = ((a' v (a ^ b)) ^ (b' v 1))
4 an1 106 . . . . 5 ((a' v (a ^ b)) ^ 1) = (a' v (a ^ b))
5 df-t 41 . . . . . . 7 1 = (a v a')
65lor 70 . . . . . 6 (b' v 1) = (b' v (a v a'))
76lan 77 . . . . 5 ((a' v (a ^ b)) ^ (b' v 1)) = ((a' v (a ^ b)) ^ (b' v (a v a')))
83, 4, 73tr2 64 . . . 4 (a' v (a ^ b)) = ((a' v (a ^ b)) ^ (b' v (a v a')))
9 ax-a2 31 . . . . . 6 (a v a') = (a' v a)
109lor 70 . . . . 5 (b' v (a v a')) = (b' v (a' v a))
1110lan 77 . . . 4 ((a' v (a ^ b)) ^ (b' v (a v a'))) = ((a' v (a ^ b)) ^ (b' v (a' v a)))
12 ax-a3 32 . . . . . 6 ((b' v a') v a) = (b' v (a' v a))
1312ax-r1 35 . . . . 5 (b' v (a' v a)) = ((b' v a') v a)
1413lan 77 . . . 4 ((a' v (a ^ b)) ^ (b' v (a' v a))) = ((a' v (a ^ b)) ^ ((b' v a') v a))
158, 11, 143tr 65 . . 3 (a' v (a ^ b)) = ((a' v (a ^ b)) ^ ((b' v a') v a))
16 ax-a2 31 . . . . 5 (b' v a') = (a' v b')
1716ax-r5 38 . . . 4 ((b' v a') v a) = ((a' v b') v a)
1817lan 77 . . 3 ((a' v (a ^ b)) ^ ((b' v a') v a)) = ((a' v (a ^ b)) ^ ((a' v b') v a))
19 oran3 93 . . . . 5 (a' v b') = (a ^ b)'
2019ax-r5 38 . . . 4 ((a' v b') v a) = ((a ^ b)' v a)
2120lan 77 . . 3 ((a' v (a ^ b)) ^ ((a' v b') v a)) = ((a' v (a ^ b)) ^ ((a ^ b)' v a))
2215, 18, 213tr 65 . 2 (a' v (a ^ b)) = ((a' v (a ^ b)) ^ ((a ^ b)' v a))
23 df-i0 43 . 2 (a ->0 (a ^ b)) = (a' v (a ^ b))
24 df-id0 49 . 2 (a ==0 (a ^ b)) = ((a' v (a ^ b)) ^ ((a ^ b)' v a))
2522, 23, 243tr1 63 1 (a ->0 (a ^ b)) = (a ==0 (a ^ b))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->0 wi0 11   ==0 wid0 17
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-i0 43  df-id0 49
This theorem is referenced by: (None)
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