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Theorem i3orlem6 557
 Description: Lemma for Kalmbach implication OR builder.
Assertion
Ref Expression
i3orlem6 ((a3 b) ∪ ((ac) →3 (bc))) = (((ab) ∩ (a3 b )) ∪ ((ac) →3 (bc)))

Proof of Theorem i3orlem6
StepHypRef Expression
1 ax-a3 32 . . 3 (((ab) ∪ (a3 b) ) ∪ ((ac) →3 (bc))) = ((ab) ∪ ((a3 b) ∪ ((ac) →3 (bc))))
21ax-r1 35 . 2 ((ab) ∪ ((a3 b) ∪ ((ac) →3 (bc)))) = (((ab) ∪ (a3 b) ) ∪ ((ac) →3 (bc)))
3 i3orlem2 553 . . . 4 (ab) ≤ ((ac) →3 (bc))
43lerr 150 . . 3 (ab) ≤ ((a3 b) ∪ ((ac) →3 (bc)))
54df-le2 131 . 2 ((ab) ∪ ((a3 b) ∪ ((ac) →3 (bc)))) = ((a3 b) ∪ ((ac) →3 (bc)))
6 oi3ai3 503 . . 3 ((ab) ∪ (a3 b) ) = ((ab) ∩ (a3 b ))
76ax-r5 38 . 2 (((ab) ∪ (a3 b) ) ∪ ((ac) →3 (bc))) = (((ab) ∩ (a3 b )) ∪ ((ac) →3 (bc)))
82, 5, 73tr2 64 1 ((a3 b) ∪ ((ac) →3 (bc))) = (((ab) ∩ (a3 b )) ∪ ((ac) →3 (bc)))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  i3orlem7  558  i3orlem8  559
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