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Theorem oa3-4lem 983
 Description: Lemma for 3-OA(4). Equivalence with substitution into 6-OA dual.
Assertion
Ref Expression
oa3-4lem (a ∩ (a ∪ (b ∩ (((ab) ∪ (ab )) ∪ (((ac) ∪ (a ∩ 1)) ∩ ((bc) ∪ (b ∩ 1))))))) = (a ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c))))))

Proof of Theorem oa3-4lem
StepHypRef Expression
1 dfb 94 . . . . . 6 (ab) = ((ab) ∪ (ab ))
2 ax-a2 31 . . . . . . . 8 (a ∪ (ac)) = ((ac) ∪ a )
3 df-i1 44 . . . . . . . 8 (a1 c) = (a ∪ (ac))
4 an1 106 . . . . . . . . 9 (a ∩ 1) = a
54lor 70 . . . . . . . 8 ((ac) ∪ (a ∩ 1)) = ((ac) ∪ a )
62, 3, 53tr1 63 . . . . . . 7 (a1 c) = ((ac) ∪ (a ∩ 1))
7 ax-a2 31 . . . . . . . 8 (b ∪ (bc)) = ((bc) ∪ b )
8 df-i1 44 . . . . . . . 8 (b1 c) = (b ∪ (bc))
9 an1 106 . . . . . . . . 9 (b ∩ 1) = b
109lor 70 . . . . . . . 8 ((bc) ∪ (b ∩ 1)) = ((bc) ∪ b )
117, 8, 103tr1 63 . . . . . . 7 (b1 c) = ((bc) ∪ (b ∩ 1))
126, 112an 79 . . . . . 6 ((a1 c) ∩ (b1 c)) = (((ac) ∪ (a ∩ 1)) ∩ ((bc) ∪ (b ∩ 1)))
131, 122or 72 . . . . 5 ((ab) ∪ ((a1 c) ∩ (b1 c))) = (((ab) ∪ (ab )) ∪ (((ac) ∪ (a ∩ 1)) ∩ ((bc) ∪ (b ∩ 1))))
1413ax-r1 35 . . . 4 (((ab) ∪ (ab )) ∪ (((ac) ∪ (a ∩ 1)) ∩ ((bc) ∪ (b ∩ 1)))) = ((ab) ∪ ((a1 c) ∩ (b1 c)))
1514lan 77 . . 3 (b ∩ (((ab) ∪ (ab )) ∪ (((ac) ∪ (a ∩ 1)) ∩ ((bc) ∪ (b ∩ 1))))) = (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c))))
1615lor 70 . 2 (a ∪ (b ∩ (((ab) ∪ (ab )) ∪ (((ac) ∪ (a ∩ 1)) ∩ ((bc) ∪ (b ∩ 1)))))) = (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))
1716lan 77 1 (a ∩ (a ∪ (b ∩ (((ab) ∪ (ab )) ∪ (((ac) ∪ (a ∩ 1)) ∩ ((bc) ∪ (b ∩ 1))))))) = (a ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c))))))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44 This theorem is referenced by:  oa3-2to4  988
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