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Theorem u12lembi 726
 Description: Sasaki/Dishkant implication and biconditional. Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 1, and j is set to 2.
Assertion
Ref Expression
u12lembi ((a1 b) ∩ (b2 a)) = (ab)

Proof of Theorem u12lembi
StepHypRef Expression
1 u1lemc1 680 . . . . 5 a C (a1 b)
21comcom 453 . . . 4 (a1 b) C a
3 lear 161 . . . . . . 7 (ba ) ≤ a
4 leo 158 . . . . . . . 8 a ≤ (a ∪ (ab))
5 df-i1 44 . . . . . . . . 9 (a1 b) = (a ∪ (ab))
65ax-r1 35 . . . . . . . 8 (a ∪ (ab)) = (a1 b)
74, 6lbtr 139 . . . . . . 7 a ≤ (a1 b)
83, 7letr 137 . . . . . 6 (ba ) ≤ (a1 b)
98lecom 180 . . . . 5 (ba ) C (a1 b)
109comcom 453 . . . 4 (a1 b) C (ba )
112, 10fh1 469 . . 3 ((a1 b) ∩ (a ∪ (ba ))) = (((a1 b) ∩ a) ∪ ((a1 b) ∩ (ba )))
12 u1lemaa 600 . . . 4 ((a1 b) ∩ a) = (ab)
13 an12 81 . . . . 5 ((a1 b) ∩ (ba )) = (b ∩ ((a1 b) ∩ a ))
14 u1lemana 605 . . . . . 6 ((a1 b) ∩ a ) = a
1514lan 77 . . . . 5 (b ∩ ((a1 b) ∩ a )) = (ba )
16 ancom 74 . . . . 5 (ba ) = (ab )
1713, 15, 163tr 65 . . . 4 ((a1 b) ∩ (ba )) = (ab )
1812, 172or 72 . . 3 (((a1 b) ∩ a) ∪ ((a1 b) ∩ (ba ))) = ((ab) ∪ (ab ))
1911, 18ax-r2 36 . 2 ((a1 b) ∩ (a ∪ (ba ))) = ((ab) ∪ (ab ))
20 df-i2 45 . . 3 (b2 a) = (a ∪ (ba ))
2120lan 77 . 2 ((a1 b) ∩ (b2 a)) = ((a1 b) ∩ (a ∪ (ba )))
22 dfb 94 . 2 (ab) = ((ab) ∪ (ab ))
2319, 21, 223tr1 63 1 ((a1 b) ∩ (b2 a)) = (ab)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ 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:  bi3  839  bi4  840
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