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Theorem lem3.3.7i4e2 1070
Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 4, and this is the second part of the equation. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem3.3.7i4e2 (a ==4 (a ^ b)) = ((a ^ b) ==4 a)
Proof of Theorem lem3.3.7i4e2
StepHypRef Expression
1 lear 161 . . . . . 6 (a ^ (a ^ b)) =< (a ^ b)
2 lea 160 . . . . . . 7 (a ^ b) =< a
3 leid 148 . . . . . . 7 (a ^ b) =< (a ^ b)
42, 3ler2an 173 . . . . . 6 (a ^ b) =< (a ^ (a ^ b))
51, 4lebi 145 . . . . 5 (a ^ (a ^ b)) = (a ^ b)
65lor 70 . . . 4 ((a ^ b)' v (a ^ (a ^ b))) = ((a ^ b)' v (a ^ b))
76lan 77 . . 3 ((a' v (a ^ b)) ^ ((a ^ b)' v (a ^ (a ^ b)))) = ((a' v (a ^ b)) ^ ((a ^ b)' v (a ^ b)))
83sklem 230 . . . 4 ((a ^ b)' v (a ^ b)) = 1
98lan 77 . . 3 ((a' v (a ^ b)) ^ ((a ^ b)' v (a ^ b))) = ((a' v (a ^ b)) ^ 1)
10 an1 106 . . . 4 ((a' v (a ^ b)) ^ 1) = (a' v (a ^ b))
112df2le2 136 . . . . . . 7 ((a ^ b) ^ a) = (a ^ b)
1211ax-r1 35 . . . . . 6 (a ^ b) = ((a ^ b) ^ a)
1312lor 70 . . . . 5 (a' v (a ^ b)) = (a' v ((a ^ b) ^ a))
14 an1r 107 . . . . . 6 (1 ^ (a' v ((a ^ b) ^ a))) = (a' v ((a ^ b) ^ a))
1514ax-r1 35 . . . . 5 (a' v ((a ^ b) ^ a)) = (1 ^ (a' v ((a ^ b) ^ a)))
1613, 15ax-r2 36 . . . 4 (a' v (a ^ b)) = (1 ^ (a' v ((a ^ b) ^ a)))
172sklem 230 . . . . . 6 ((a ^ b)' v a) = 1
1817ax-r1 35 . . . . 5 1 = ((a ^ b)' v a)
1918ran 78 . . . 4 (1 ^ (a' v ((a ^ b) ^ a))) = (((a ^ b)' v a) ^ (a' v ((a ^ b) ^ a)))
2010, 16, 193tr 65 . . 3 ((a' v (a ^ b)) ^ 1) = (((a ^ b)' v a) ^ (a' v ((a ^ b) ^ a)))
217, 9, 203tr 65 . 2 ((a' v (a ^ b)) ^ ((a ^ b)' v (a ^ (a ^ b)))) = (((a ^ b)' v a) ^ (a' v ((a ^ b) ^ a)))
22 df-id4 53 . 2 (a ==4 (a ^ b)) = ((a' v (a ^ b)) ^ ((a ^ b)' v (a ^ (a ^ b))))
23 df-id4 53 . 2 ((a ^ b) ==4 a) = (((a ^ b)' v a) ^ (a' v ((a ^ b) ^ a)))
2421, 22, 233tr1 63 1 (a ==4 (a ^ b)) = ((a ^ b) ==4 a)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ==4 wid4 21
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-id4 53  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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