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Theorem oa3-2to4 988
 Description: Derivation of 3-OA variant (4) from (2).
Hypothesis
Ref Expression
oa3-2to4.1 ((a1 c) ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))) ≤ c
Assertion
Ref Expression
oa3-2to4 (a ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))) ≤ c

Proof of Theorem oa3-2to4
StepHypRef Expression
1 oa3-4lem 983 . . 3 (a ∩ (a ∪ (b ∩ (((ab) ∪ (ab )) ∪ (((ac) ∪ (a ∩ 1)) ∩ ((bc) ∪ (b ∩ 1))))))) = (a ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c))))))
21ax-r1 35 . 2 (a ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))) = (a ∩ (a ∪ (b ∩ (((ab) ∪ (ab )) ∪ (((ac) ∪ (a ∩ 1)) ∩ ((bc) ∪ (b ∩ 1)))))))
3 leid 148 . . 3 aa
4 leid 148 . . 3 bb
5 le1 146 . . 3 c ≤ 1
6 an1 106 . . . . . . 7 (c ∩ 1) = c
7 dff 101 . . . . . . . . . 10 0 = (aa )
8 dff 101 . . . . . . . . . 10 0 = (bb )
97, 82or 72 . . . . . . . . 9 (0 ∪ 0) = ((aa ) ∪ (bb ))
109ax-r1 35 . . . . . . . 8 ((aa ) ∪ (bb )) = (0 ∪ 0)
11 or0 102 . . . . . . . 8 (0 ∪ 0) = 0
1210, 11ax-r2 36 . . . . . . 7 ((aa ) ∪ (bb )) = 0
136, 122or 72 . . . . . 6 ((c ∩ 1) ∪ ((aa ) ∪ (bb ))) = (c ∪ 0)
14 or0 102 . . . . . 6 (c ∪ 0) = c
1513, 14ax-r2 36 . . . . 5 ((c ∩ 1) ∪ ((aa ) ∪ (bb ))) = c
1615ax-r1 35 . . . 4 c = ((c ∩ 1) ∪ ((aa ) ∪ (bb )))
17 ax-a2 31 . . . 4 ((c ∩ 1) ∪ ((aa ) ∪ (bb ))) = (((aa ) ∪ (bb )) ∪ (c ∩ 1))
1816, 17ax-r2 36 . . 3 c = (((aa ) ∪ (bb )) ∪ (c ∩ 1))
19 oa3-2lemb 979 . . . 4 ((a1 c) ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((ac) ∪ ((a1 c) ∩ (c1 c))) ∩ ((bc) ∪ ((b1 c) ∩ (c1 c)))))))) = ((a1 c) ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c))))))
20 oa3-2to4.1 . . . 4 ((a1 c) ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))) ≤ c
2119, 20bltr 138 . . 3 ((a1 c) ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((ac) ∪ ((a1 c) ∩ (c1 c))) ∩ ((bc) ∪ ((b1 c) ∩ (c1 c)))))))) ≤ c
223, 4, 5, 18, 21oa4to6dual 964 . 2 (a ∩ (a ∪ (b ∩ (((ab) ∪ (ab )) ∪ (((ac) ∪ (a ∩ 1)) ∩ ((bc) ∪ (b ∩ 1))))))) ≤ c
232, 22bltr 138 1 (a ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))) ≤ c
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →1 wi1 12 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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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