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Theorem u2lembi 721
 Description: Dishkant implication and biconditional.
Assertion
Ref Expression
u2lembi ((a2 b) ∩ (b2 a)) = (ab)

Proof of Theorem u2lembi
StepHypRef Expression
1 ancom 74 . . 3 ((b ∪ (ab )) ∩ (a ∪ (ab ))) = ((a ∪ (ab )) ∩ (b ∪ (ab )))
2 coman1 185 . . . . . 6 (ab ) C a
32comcom7 460 . . . . 5 (ab ) C a
4 coman2 186 . . . . . 6 (ab ) C b
54comcom7 460 . . . . 5 (ab ) C b
63, 5fh3r 475 . . . 4 ((ab) ∪ (ab )) = ((a ∪ (ab )) ∩ (b ∪ (ab )))
76ax-r1 35 . . 3 ((a ∪ (ab )) ∩ (b ∪ (ab ))) = ((ab) ∪ (ab ))
81, 7ax-r2 36 . 2 ((b ∪ (ab )) ∩ (a ∪ (ab ))) = ((ab) ∪ (ab ))
9 df-i2 45 . . 3 (a2 b) = (b ∪ (ab ))
10 df-i2 45 . . . 4 (b2 a) = (a ∪ (ba ))
11 ancom 74 . . . . 5 (ba ) = (ab )
1211lor 70 . . . 4 (a ∪ (ba )) = (a ∪ (ab ))
1310, 12ax-r2 36 . . 3 (b2 a) = (a ∪ (ab ))
149, 132an 79 . 2 ((a2 b) ∩ (b2 a)) = ((b ∪ (ab )) ∩ (a ∪ (ab )))
15 dfb 94 . 2 (ab) = ((ab) ∪ (ab ))
168, 14, 153tr1 63 1 ((a2 b) ∩ (b2 a)) = (ab)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  i2bi  722  mloa  1018
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