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Theorem wdid0id4 1113
 Description: Show a quantum identity that follows from classical identity in a WDOL.
Hypothesis
Ref Expression
wdid0id5.1 (a0 b) = 1
Assertion
Ref Expression
wdid0id4 (a4 b) = 1

Proof of Theorem wdid0id4
StepHypRef Expression
1 df-id4 53 . 2 (a4 b) = ((ab) ∩ (b ∪ (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 wddi3 1107 . . . . . 6 ((b ∪ (ab)) ≡ ((ba) ∩ (bb))) = 1
7 wa2 192 . . . . . . . 8 ((bb) ≡ (bb )) = 1
87wlan 370 . . . . . . 7 (((ba) ∩ (bb)) ≡ ((ba) ∩ (bb ))) = 1
9 wa4 194 . . . . . . . 8 (((ba) ∪ (bb )) ≡ (bb )) = 1
109wleoa 376 . . . . . . 7 (((ba) ∩ (bb )) ≡ (ba)) = 1
118, 10wr2 371 . . . . . 6 (((ba) ∩ (bb)) ≡ (ba)) = 1
126, 11wr2 371 . . . . 5 ((b ∪ (ab)) ≡ (ba)) = 1
1312wr1 197 . . . 4 ((ba) ≡ (b ∪ (ab))) = 1
1413wlan 370 . . 3 (((ab) ∩ (ba)) ≡ ((ab) ∩ (b ∪ (ab)))) = 1
155, 14wwbmp 205 . 2 ((ab) ∩ (b ∪ (ab))) = 1
161, 15ax-r2 36 1 (a4 b) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   ≡0 wid0 17   ≡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  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-id4 53  df-le 129  df-le1 130  df-le2 131  df-cmtr 134 This theorem is referenced by:  wddi-4  1119
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