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Theorem wdid0id1 1110
Description: Show a quantum identity that follows from classical identity in a WDOL.
Hypothesis
Ref Expression
wdid0id5.1 (a ==0 b) = 1
Assertion
Ref Expression
wdid0id1 (a ==1 b) = 1
Proof of Theorem wdid0id1
StepHypRef Expression
1 df-id1 50 . 2 (a ==1 b) = ((a v b') ^ (a' 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 wancom 203 . . . . . . . 8 (((a' v a) ^ (a' v b)) == ((a' v b) ^ (a' v a))) = 1
7 wa2 192 . . . . . . . . . 10 ((a' v a) == (a v a')) = 1
87wlan 370 . . . . . . . . 9 (((a' v b) ^ (a' v a)) == ((a' v b) ^ (a v a'))) = 1
9 wa4 194 . . . . . . . . . 10 (((a' v b) v (a v a')) == (a v a')) = 1
109wleoa 376 . . . . . . . . 9 (((a' v b) ^ (a v a')) == (a' v b)) = 1
118, 10wr2 371 . . . . . . . 8 (((a' v b) ^ (a' v a)) == (a' v b)) = 1
126, 11wr2 371 . . . . . . 7 (((a' v a) ^ (a' v b)) == (a' v b)) = 1
1312wr1 197 . . . . . 6 ((a' v b) == ((a' v a) ^ (a' v b))) = 1
14 wddi3 1107 . . . . . . 7 ((a' v (a ^ b)) == ((a' v a) ^ (a' v b))) = 1
1514wr1 197 . . . . . 6 (((a' v a) ^ (a' v b)) == (a' v (a ^ b))) = 1
1613, 15wr2 371 . . . . 5 ((a' v b) == (a' v (a ^ b))) = 1
17 wa2 192 . . . . 5 ((b' v a) == (a v b')) = 1
1816, 17w2an 373 . . . 4 (((a' v b) ^ (b' v a)) == ((a' v (a ^ b)) ^ (a v b'))) = 1
19 biid 116 . . . 4 (((a' v b) ^ (b' v a)) == ((a' v b) ^ (b' v a))) = 1
20 wancom 203 . . . 4 (((a v b') ^ (a' v (a ^ b))) == ((a' v (a ^ b)) ^ (a v b'))) = 1
2118, 19, 20w3tr1 374 . . 3 (((a' v b) ^ (b' v a)) == ((a v b') ^ (a' v (a ^ b)))) = 1
225, 21wwbmp 205 . 2 ((a v b') ^ (a' v (a ^ b))) = 1
231, 22ax-r2 36 1 (a ==1 b) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ==0 wid0 17   ==1 wid1 18
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-id1 50  df-le 129  df-le1 130  df-le2 131  df-cmtr 134
This theorem is referenced by:  wddi-1  1116
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