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Theorem wql2lem2 289
 Description: Lemma for →2 WQL axiom. (Contributed by NM, 6-Dec-1998.)
Hypothesis
Ref Expression
wql2lem2.1 ((ac) →2 (bc)) = 1
Assertion
Ref Expression
wql2lem2 ((a ∪ (bc)) ∪ (bc)) = 1

Proof of Theorem wql2lem2
StepHypRef Expression
1 df-i2 45 . . . 4 ((ac) →2 (bc)) = ((bc) ∪ ((ac) ∩ (bc) ))
2 anor3 90 . . . . . 6 ((ac) ∩ (bc) ) = ((ac) ∪ (bc))
3 ax-a3 32 . . . . . . . . . 10 ((ab) ∪ c) = (a ∪ (bc))
43ax-r1 35 . . . . . . . . 9 (a ∪ (bc)) = ((ab) ∪ c)
5 orordir 113 . . . . . . . . 9 ((ab) ∪ c) = ((ac) ∪ (bc))
64, 5ax-r2 36 . . . . . . . 8 (a ∪ (bc)) = ((ac) ∪ (bc))
76ax-r4 37 . . . . . . 7 (a ∪ (bc)) = ((ac) ∪ (bc))
87ax-r1 35 . . . . . 6 ((ac) ∪ (bc)) = (a ∪ (bc))
92, 8ax-r2 36 . . . . 5 ((ac) ∩ (bc) ) = (a ∪ (bc))
109lor 70 . . . 4 ((bc) ∪ ((ac) ∩ (bc) )) = ((bc) ∪ (a ∪ (bc)) )
11 ax-a2 31 . . . 4 ((bc) ∪ (a ∪ (bc)) ) = ((a ∪ (bc)) ∪ (bc))
121, 10, 113tr 65 . . 3 ((ac) →2 (bc)) = ((a ∪ (bc)) ∪ (bc))
1312ax-r1 35 . 2 ((a ∪ (bc)) ∪ (bc)) = ((ac) →2 (bc))
14 wql2lem2.1 . 2 ((ac) →2 (bc)) = 1
1513, 14ax-r2 36 1 ((a ∪ (bc)) ∪ (bc)) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →2 wi2 13 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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 This theorem is referenced by:  wql2lem4  291
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