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Theorem i2i1i1 800
 Description: Equivalence to →2 .
Assertion
Ref Expression
i2i1i1 (a2 b) = ((a1 (ab)) ∩ ((ab) →1 b))

Proof of Theorem i2i1i1
StepHypRef Expression
1 an1r 107 . . 3 (1 ∩ (b ∪ (ab ))) = (b ∪ (ab ))
21ax-r1 35 . 2 (b ∪ (ab )) = (1 ∩ (b ∪ (ab )))
3 df-i2 45 . 2 (a2 b) = (b ∪ (ab ))
4 anabs 121 . . . . . 6 (a ∩ (ab)) = a
54lor 70 . . . . 5 (a ∪ (a ∩ (ab))) = (aa)
6 ax-a2 31 . . . . 5 (aa) = (aa )
75, 6ax-r2 36 . . . 4 (a ∪ (a ∩ (ab))) = (aa )
8 df-i1 44 . . . 4 (a1 (ab)) = (a ∪ (a ∩ (ab)))
9 df-t 41 . . . 4 1 = (aa )
107, 8, 93tr1 63 . . 3 (a1 (ab)) = 1
11 df-i1 44 . . . 4 ((ab) →1 b) = ((ab) ∪ ((ab) ∩ b))
12 anor3 90 . . . . . 6 (ab ) = (ab)
13 leor 159 . . . . . . . 8 b ≤ (ab)
14 leid 148 . . . . . . . 8 bb
1513, 14ler2an 173 . . . . . . 7 b ≤ ((ab) ∩ b)
16 lear 161 . . . . . . 7 ((ab) ∩ b) ≤ b
1715, 16lebi 145 . . . . . 6 b = ((ab) ∩ b)
1812, 172or 72 . . . . 5 ((ab ) ∪ b) = ((ab) ∪ ((ab) ∩ b))
1918ax-r1 35 . . . 4 ((ab) ∪ ((ab) ∩ b)) = ((ab ) ∪ b)
20 ax-a2 31 . . . 4 ((ab ) ∪ b) = (b ∪ (ab ))
2111, 19, 203tr 65 . . 3 ((ab) →1 b) = (b ∪ (ab ))
2210, 212an 79 . 2 ((a1 (ab)) ∩ ((ab) →1 b)) = (1 ∩ (b ∪ (ab )))
232, 3, 223tr1 63 1 (a2 b) = ((a1 (ab)) ∩ ((ab) →1 b))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12   →2 wi2 13 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by:  mlaconj  845
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