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Theorem u4lem2 747
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u4lem2 (((a4 b) →4 a) →4 a) = (a ∪ ((ab) ∪ (ab )))

Proof of Theorem u4lem2
StepHypRef Expression
1 u4lemc1 683 . . . 4 a C ((a4 b) →4 a)
21comcom 453 . . 3 ((a4 b) →4 a) C a
32u4lemc4 704 . 2 (((a4 b) →4 a) →4 a) = (((a4 b) →4 a)a)
4 u4lem1n 742 . . . 4 ((a4 b) →4 a) = ((((ab) ∩ (ab )) ∩ a) ∪ ((ab) ∪ (ab )))
54ax-r5 38 . . 3 (((a4 b) →4 a)a) = (((((ab) ∩ (ab )) ∩ a) ∪ ((ab) ∪ (ab ))) ∪ a)
6 ax-a3 32 . . . 4 (((((ab) ∩ (ab )) ∩ a) ∪ ((ab) ∪ (ab ))) ∪ a) = ((((ab) ∩ (ab )) ∩ a) ∪ (((ab) ∪ (ab )) ∪ a))
7 lear 161 . . . . . . 7 (((ab) ∩ (ab )) ∩ a) ≤ a
8 leor 159 . . . . . . 7 a ≤ (((ab) ∪ (ab )) ∪ a)
97, 8letr 137 . . . . . 6 (((ab) ∩ (ab )) ∩ a) ≤ (((ab) ∪ (ab )) ∪ a)
109df-le2 131 . . . . 5 ((((ab) ∩ (ab )) ∩ a) ∪ (((ab) ∪ (ab )) ∪ a)) = (((ab) ∪ (ab )) ∪ a)
11 ax-a2 31 . . . . 5 (((ab) ∪ (ab )) ∪ a) = (a ∪ ((ab) ∪ (ab )))
1210, 11ax-r2 36 . . . 4 ((((ab) ∩ (ab )) ∩ a) ∪ (((ab) ∪ (ab )) ∪ a)) = (a ∪ ((ab) ∪ (ab )))
136, 12ax-r2 36 . . 3 (((((ab) ∩ (ab )) ∩ a) ∪ ((ab) ∪ (ab ))) ∪ a) = (a ∪ ((ab) ∪ (ab )))
145, 13ax-r2 36 . 2 (((a4 b) →4 a)a) = (a ∪ ((ab) ∪ (ab )))
153, 14ax-r2 36 1 (((a4 b) →4 a) →4 a) = (a ∪ ((ab) ∪ (ab )))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →4 wi4 15 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-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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