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Theorem lem3.3.7i2e1 1063
Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 2, and this is the first part of the equation. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem3.3.7i2e1 (a ->2 (a ^ b)) = (a ==2 (a ^ b))
Proof of Theorem lem3.3.7i2e1
StepHypRef Expression
1 or1r 105 . . . . . 6 (1 v b') = 1
21ax-r1 35 . . . . 5 1 = (1 v b')
32ran 78 . . . 4 (1 ^ ((a ^ b) v (a' ^ (a ^ b)'))) = ((1 v b') ^ ((a ^ b) v (a' ^ (a ^ b)')))
4 an1r 107 . . . 4 (1 ^ ((a ^ b) v (a' ^ (a ^ b)'))) = ((a ^ b) v (a' ^ (a ^ b)'))
5 df-t 41 . . . . . 6 1 = (a v a')
65ax-r5 38 . . . . 5 (1 v b') = ((a v a') v b')
76ran 78 . . . 4 ((1 v b') ^ ((a ^ b) v (a' ^ (a ^ b)'))) = (((a v a') v b') ^ ((a ^ b) v (a' ^ (a ^ b)')))
83, 4, 73tr2 64 . . 3 ((a ^ b) v (a' ^ (a ^ b)')) = (((a v a') v b') ^ ((a ^ b) v (a' ^ (a ^ b)')))
9 ax-a3 32 . . . 4 ((a v a') v b') = (a v (a' v b'))
109ran 78 . . 3 (((a v a') v b') ^ ((a ^ b) v (a' ^ (a ^ b)'))) = ((a v (a' v b')) ^ ((a ^ b) v (a' ^ (a ^ b)')))
11 oran3 93 . . . . 5 (a' v b') = (a ^ b)'
1211lor 70 . . . 4 (a v (a' v b')) = (a v (a ^ b)')
1312ran 78 . . 3 ((a v (a' v b')) ^ ((a ^ b) v (a' ^ (a ^ b)'))) = ((a v (a ^ b)') ^ ((a ^ b) v (a' ^ (a ^ b)')))
148, 10, 133tr 65 . 2 ((a ^ b) v (a' ^ (a ^ b)')) = ((a v (a ^ b)') ^ ((a ^ b) v (a' ^ (a ^ b)')))
15 df-i2 45 . 2 (a ->2 (a ^ b)) = ((a ^ b) v (a' ^ (a ^ b)'))
16 df-id2 51 . 2 (a ==2 (a ^ b)) = ((a v (a ^ b)') ^ ((a ^ b) v (a' ^ (a ^ b)')))
1714, 15, 163tr1 63 1 (a ->2 (a ^ b)) = (a ==2 (a ^ b))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->2 wi2 13   ==2 wid2 19
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-i2 45  df-id2 51
This theorem is referenced by: (None)
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