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Theorem ud4lem3a 583
 Description: Lemma for unified disjunction.
Assertion
Ref Expression
ud4lem3a ((a4 b) ∩ (ab)) = (a4 b)

Proof of Theorem ud4lem3a
StepHypRef Expression
1 anass 76 . . 3 ((((ab ) ∩ (ab )) ∩ ((ab ) ∪ b)) ∩ (ab)) = (((ab ) ∩ (ab )) ∩ (((ab ) ∪ b) ∩ (ab)))
2 lea 160 . . . . . 6 (ab ) ≤ a
32leror 152 . . . . 5 ((ab ) ∪ b) ≤ (ab)
43df2le2 136 . . . 4 (((ab ) ∪ b) ∩ (ab)) = ((ab ) ∪ b)
54lan 77 . . 3 (((ab ) ∩ (ab )) ∩ (((ab ) ∪ b) ∩ (ab))) = (((ab ) ∩ (ab )) ∩ ((ab ) ∪ b))
61, 5ax-r2 36 . 2 ((((ab ) ∩ (ab )) ∩ ((ab ) ∪ b)) ∩ (ab)) = (((ab ) ∩ (ab )) ∩ ((ab ) ∪ b))
7 ud4lem0c 280 . . 3 (a4 b) = (((ab ) ∩ (ab )) ∩ ((ab ) ∪ b))
87ran 78 . 2 ((a4 b) ∩ (ab)) = ((((ab ) ∩ (ab )) ∩ ((ab ) ∪ b)) ∩ (ab))
96, 8, 73tr1 63 1 ((a4 b) ∩ (ab)) = (a4 b)
 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-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-i4 47  df-le1 130  df-le2 131 This theorem is referenced by:  ud4lem3  585
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