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Theorem wdid0id5 1109
Description: Show that quantum identity follows from classical identity in a WDOL.
Hypothesis
Ref Expression
wdid0id5.1 (a ==0 b) = 1
Assertion
Ref Expression
wdid0id5 (a == b) = 1
Proof of Theorem wdid0id5
StepHypRef Expression
1 dfb 94 . 2 (a == b) = ((a ^ b) v (a' ^ b'))
2 df-id0 49 . . . . 5 (a ==0 b) = ((a' v b) ^ (b' v a))
32ax-r1 35 . . . 4 ((a' v b) ^ (b' v a)) = (a ==0 b)
4 wdid0id5.1 . . . 4 (a ==0 b) = 1
53, 4ax-r2 36 . . 3 ((a' v b) ^ (b' v a)) = 1
6 wa4 194 . . . . . . . . 9 (((a v b') v (a v a')) == (a v a')) = 1
76wleoa 376 . . . . . . . 8 (((a v b') ^ (a v a')) == (a v b')) = 1
87wr1 197 . . . . . . 7 ((a v b') == ((a v b') ^ (a v a'))) = 1
9 wancom 203 . . . . . . 7 (((a v b') ^ (a v a')) == ((a v a') ^ (a v b'))) = 1
108, 9wr2 371 . . . . . 6 ((a v b') == ((a v a') ^ (a v b'))) = 1
11 wa2 192 . . . . . 6 ((b' v a) == (a v b')) = 1
12 wddi3 1107 . . . . . 6 ((a v (a' ^ b')) == ((a v a') ^ (a v b'))) = 1
1310, 11, 12w3tr1 374 . . . . 5 ((b' v a) == (a v (a' ^ b'))) = 1
14 wa4 194 . . . . . . . 8 (((b v a') v (b v b')) == (b v b')) = 1
1514wleoa 376 . . . . . . 7 (((b v a') ^ (b v b')) == (b v a')) = 1
1615wr1 197 . . . . . 6 ((b v a') == ((b v a') ^ (b v b'))) = 1
17 wa2 192 . . . . . 6 ((a' v b) == (b v a')) = 1
18 wddi3 1107 . . . . . 6 ((b v (a' ^ b')) == ((b v a') ^ (b v b'))) = 1
1916, 17, 18w3tr1 374 . . . . 5 ((a' v b) == (b v (a' ^ b'))) = 1
2013, 19w2an 373 . . . 4 (((b' v a) ^ (a' v b)) == ((a v (a' ^ b')) ^ (b v (a' ^ b')))) = 1
21 wancom 203 . . . 4 (((a' v b) ^ (b' v a)) == ((b' v a) ^ (a' v b))) = 1
22 wddi4 1108 . . . 4 (((a ^ b) v (a' ^ b')) == ((a v (a' ^ b')) ^ (b v (a' ^ b')))) = 1
2320, 21, 22w3tr1 374 . . 3 (((a' v b) ^ (b' v a)) == ((a ^ b) v (a' ^ b'))) = 1
245, 23wwbmp 205 . 2 ((a ^ b) v (a' ^ b')) = 1
251, 24ax-r2 36 1 (a == b) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8   ==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  ax-wom 361  ax-wdol 1102
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-id0 49  df-le 129  df-le1 130  df-le2 131  df-cmtr 134
This theorem is referenced by:  wdka4o  1114
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