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Theorem u2lemona 626
 Description: Lemma for Dishkant implication study.
Assertion
Ref Expression
u2lemona ((a2 b) ∪ a ) = (ab)

Proof of Theorem u2lemona
StepHypRef Expression
1 df-i2 45 . . 3 (a2 b) = (b ∪ (ab ))
21ax-r5 38 . 2 ((a2 b) ∪ a ) = ((b ∪ (ab )) ∪ a )
3 ax-a3 32 . . 3 ((b ∪ (ab )) ∪ a ) = (b ∪ ((ab ) ∪ a ))
4 ax-a2 31 . . . 4 (b ∪ ((ab ) ∪ a )) = (((ab ) ∪ a ) ∪ b)
5 lea 160 . . . . . 6 (ab ) ≤ a
65df-le2 131 . . . . 5 ((ab ) ∪ a ) = a
76ax-r5 38 . . . 4 (((ab ) ∪ a ) ∪ b) = (ab)
84, 7ax-r2 36 . . 3 (b ∪ ((ab ) ∪ a )) = (ab)
93, 8ax-r2 36 . 2 ((b ∪ (ab )) ∪ a ) = (ab)
102, 9ax-r2 36 1 ((a2 b) ∪ a ) = (ab)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →2 wi2 13 This theorem was proved from axioms:  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r5 38 This theorem depends on definitions:  df-a 40  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by:  u2lemnaa  641
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