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Theorem u1lembi 720
 Description: Sasaki implication and biconditional.
Assertion
Ref Expression
u1lembi ((a1 b) ∩ (b1 a)) = (ab)

Proof of Theorem u1lembi
StepHypRef Expression
1 ax-a2 31 . . . 4 (a ∪ (ab)) = ((ab) ∪ a )
2 ax-a2 31 . . . 4 (b ∪ (ab)) = ((ab) ∪ b )
31, 22an 79 . . 3 ((a ∪ (ab)) ∩ (b ∪ (ab))) = (((ab) ∪ a ) ∩ ((ab) ∪ b ))
4 coman1 185 . . . . . 6 (ab) C a
54comcom2 183 . . . . 5 (ab) C a
6 coman2 186 . . . . . 6 (ab) C b
76comcom2 183 . . . . 5 (ab) C b
85, 7fh3 471 . . . 4 ((ab) ∪ (ab )) = (((ab) ∪ a ) ∩ ((ab) ∪ b ))
98ax-r1 35 . . 3 (((ab) ∪ a ) ∩ ((ab) ∪ b )) = ((ab) ∪ (ab ))
103, 9ax-r2 36 . 2 ((a ∪ (ab)) ∩ (b ∪ (ab))) = ((ab) ∪ (ab ))
11 df-i1 44 . . 3 (a1 b) = (a ∪ (ab))
12 df-i1 44 . . . 4 (b1 a) = (b ∪ (ba))
13 ancom 74 . . . . 5 (ba) = (ab)
1413lor 70 . . . 4 (b ∪ (ba)) = (b ∪ (ab))
1512, 14ax-r2 36 . . 3 (b1 a) = (b ∪ (ab))
1611, 152an 79 . 2 ((a1 b) ∩ (b1 a)) = ((a ∪ (ab)) ∩ (b ∪ (ab)))
17 dfb 94 . 2 (ab) = ((ab) ∪ (ab ))
1810, 16, 173tr1 63 1 ((a1 b) ∩ (b1 a)) = (ab)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ 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 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:  mlaoml  833  comanblem1  870
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