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Theorem 1oath1i1u 828
 Description: Orthoarguesian-like OM law. (Contributed by NM, 28-Feb-1999.)
Assertion
Ref Expression
1oath1i1u ((a1 c) ∩ (((a1 c) ∩ (b1 c))1 ((a1 c) ∩ (b1 c)))) = ((a1 c) ∩ (b1 c))

Proof of Theorem 1oath1i1u
StepHypRef Expression
1 2oath1i1 827 . 2 (((a1 c) →1 c) ∩ (((a1 c) ∩ (b1 c))2 (((a1 c) →1 c) ∩ ((b1 c) →1 c)))) = (((a1 c) →1 c) ∩ ((b1 c) →1 c))
2 u1lem11 780 . . 3 ((a1 c) →1 c) = (a1 c)
3 u1lem11 780 . . . . . 6 ((b1 c) →1 c) = (b1 c)
42, 32an 79 . . . . 5 (((a1 c) →1 c) ∩ ((b1 c) →1 c)) = ((a1 c) ∩ (b1 c))
54ud2lem0a 258 . . . 4 (((a1 c) ∩ (b1 c))2 (((a1 c) →1 c) ∩ ((b1 c) →1 c))) = (((a1 c) ∩ (b1 c))2 ((a1 c) ∩ (b1 c)))
6 i1i2con2 269 . . . . 5 (((a1 c) ∩ (b1 c))1 ((a1 c) ∩ (b1 c))) = (((a1 c) ∩ (b1 c))2 ((a1 c) ∩ (b1 c)))
76ax-r1 35 . . . 4 (((a1 c) ∩ (b1 c))2 ((a1 c) ∩ (b1 c))) = (((a1 c) ∩ (b1 c))1 ((a1 c) ∩ (b1 c)))
85, 7ax-r2 36 . . 3 (((a1 c) ∩ (b1 c))2 (((a1 c) →1 c) ∩ ((b1 c) →1 c))) = (((a1 c) ∩ (b1 c))1 ((a1 c) ∩ (b1 c)))
92, 82an 79 . 2 (((a1 c) →1 c) ∩ (((a1 c) ∩ (b1 c))2 (((a1 c) →1 c) ∩ ((b1 c) →1 c)))) = ((a1 c) ∩ (((a1 c) ∩ (b1 c))1 ((a1 c) ∩ (b1 c))))
101, 9, 43tr2 64 1 ((a1 c) ∩ (((a1 c) ∩ (b1 c))1 ((a1 c) ∩ (b1 c)))) = ((a1 c) ∩ (b1 c))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∩ wa 7   →1 wi1 12   →2 wi2 13 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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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