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Theorem 3oa2 1024
 Description: Alternate form for the 3-variable orthoarguesion law.
Assertion
Ref Expression
3oa2 ((a1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))) ≤ (b1 c)

Proof of Theorem 3oa2
StepHypRef Expression
1 ax-3oa 998 . 2 (((a1 c) →1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (((a1 c) →1 c) ∩ ((b1 c) →1 c)))) ≤ ((b1 c) →1 c)
2 u1lem11 780 . . 3 ((a1 c) →1 c) = (a1 c)
3 ax-a2 31 . . . 4 (((a1 c) ∩ (b1 c)) ∪ (((a1 c) →1 c) ∩ ((b1 c) →1 c))) = ((((a1 c) →1 c) ∩ ((b1 c) →1 c)) ∪ ((a1 c) ∩ (b1 c)))
4 u1lem11 780 . . . . . 6 ((b1 c) →1 c) = (b1 c)
52, 42an 79 . . . . 5 (((a1 c) →1 c) ∩ ((b1 c) →1 c)) = ((a1 c) ∩ (b1 c))
65ax-r5 38 . . . 4 ((((a1 c) →1 c) ∩ ((b1 c) →1 c)) ∪ ((a1 c) ∩ (b1 c))) = (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))
73, 6ax-r2 36 . . 3 (((a1 c) ∩ (b1 c)) ∪ (((a1 c) →1 c) ∩ ((b1 c) →1 c))) = (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))
82, 72an 79 . 2 (((a1 c) →1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ (((a1 c) →1 c) ∩ ((b1 c) →1 c)))) = ((a1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))))
91, 8, 4le3tr2 141 1 ((a1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))) ≤ (b1 c)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →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  ax-3oa 998 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:  3oa3  1025
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