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Theorem wdid0id3 1112
 Description: Show a quantum identity that follows from classical identity in a WDOL.
Hypothesis
Ref Expression
wdid0id5.1 (a0 b) = 1
Assertion
Ref Expression
wdid0id3 (a3 b) = 1

Proof of Theorem wdid0id3
StepHypRef Expression
1 df-id3 52 . 2 (a3 b) = ((ab) ∩ (a ∪ (ab )))
2 df-id0 49 . . . . 5 (a0 b) = ((ab) ∩ (ba))
32ax-r1 35 . . . 4 ((ab) ∩ (ba)) = (a0 b)
4 wdid0id5.1 . . . 4 (a0 b) = 1
53, 4ax-r2 36 . . 3 ((ab) ∩ (ba)) = 1
6 wa4 194 . . . . . . . 8 (((ab ) ∪ (aa )) ≡ (aa )) = 1
76wleoa 376 . . . . . . 7 (((ab ) ∩ (aa )) ≡ (ab )) = 1
87wr1 197 . . . . . 6 ((ab ) ≡ ((ab ) ∩ (aa ))) = 1
9 wancom 203 . . . . . 6 (((ab ) ∩ (aa )) ≡ ((aa ) ∩ (ab ))) = 1
108, 9wr2 371 . . . . 5 ((ab ) ≡ ((aa ) ∩ (ab ))) = 1
11 wa2 192 . . . . 5 ((ba) ≡ (ab )) = 1
12 wddi3 1107 . . . . 5 ((a ∪ (ab )) ≡ ((aa ) ∩ (ab ))) = 1
1310, 11, 12w3tr1 374 . . . 4 ((ba) ≡ (a ∪ (ab ))) = 1
1413wlan 370 . . 3 (((ab) ∩ (ba)) ≡ ((ab) ∩ (a ∪ (ab )))) = 1
155, 14wwbmp 205 . 2 ((ab) ∩ (a ∪ (ab ))) = 1
161, 15ax-r2 36 1 (a3 b) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   ≡0 wid0 17   ≡3 wid3 20 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-id3 52  df-le 129  df-le1 130  df-le2 131  df-cmtr 134 This theorem is referenced by:  wddi-3  1118
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