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Theorem bi1o1a 798
Description: Equivalence to biconditional. (Contributed by NM, 5-Jul-2000.)
Assertion
Ref Expression
bi1o1a (ab) = ((a1 (ab)) ∩ ((ab) →1 a))

Proof of Theorem bi1o1a
StepHypRef Expression
1 lea 160 . . . . . . 7 (ab ) ≤ a
2 leo 158 . . . . . . 7 a ≤ (a ∪ (ab))
31, 2letr 137 . . . . . 6 (ab ) ≤ (a ∪ (ab))
43lecom 180 . . . . 5 (ab ) C (a ∪ (ab))
54comcom 453 . . . 4 (a ∪ (ab)) C (ab )
6 comor1 461 . . . . 5 (a ∪ (ab)) C a
76comcom7 460 . . . 4 (a ∪ (ab)) C a
85, 7fh1 469 . . 3 ((a ∪ (ab)) ∩ ((ab ) ∪ a)) = (((a ∪ (ab)) ∩ (ab )) ∪ ((a ∪ (ab)) ∩ a))
98ax-r1 35 . 2 (((a ∪ (ab)) ∩ (ab )) ∪ ((a ∪ (ab)) ∩ a)) = ((a ∪ (ab)) ∩ ((ab ) ∪ a))
10 dfb 94 . . 3 (ab) = ((ab) ∪ (ab ))
11 ax-a2 31 . . 3 ((ab) ∪ (ab )) = ((ab ) ∪ (ab))
12 leid 148 . . . . . 6 (ab ) ≤ (ab )
133, 12ler2an 173 . . . . 5 (ab ) ≤ ((a ∪ (ab)) ∩ (ab ))
14 lear 161 . . . . 5 ((a ∪ (ab)) ∩ (ab )) ≤ (ab )
1513, 14lebi 145 . . . 4 (ab ) = ((a ∪ (ab)) ∩ (ab ))
16 dff 101 . . . . . . 7 0 = (aa )
17 ancom 74 . . . . . . 7 (aa ) = (aa)
1816, 17ax-r2 36 . . . . . 6 0 = (aa)
1918ax-r5 38 . . . . 5 (0 ∪ ((ab) ∩ a)) = ((aa) ∪ ((ab) ∩ a))
20 lea 160 . . . . . . . 8 (ab) ≤ a
2120df2le2 136 . . . . . . 7 ((ab) ∩ a) = (ab)
2221ax-r1 35 . . . . . 6 (ab) = ((ab) ∩ a)
23 or0r 103 . . . . . . 7 (0 ∪ ((ab) ∩ a)) = ((ab) ∩ a)
2423ax-r1 35 . . . . . 6 ((ab) ∩ a) = (0 ∪ ((ab) ∩ a))
2522, 24ax-r2 36 . . . . 5 (ab) = (0 ∪ ((ab) ∩ a))
26 comid 187 . . . . . . 7 a C a
2726comcom2 183 . . . . . 6 a C a
28 comanr1 464 . . . . . 6 a C (ab)
2927, 28fh1r 473 . . . . 5 ((a ∪ (ab)) ∩ a) = ((aa) ∪ ((ab) ∩ a))
3019, 25, 293tr1 63 . . . 4 (ab) = ((a ∪ (ab)) ∩ a)
3115, 302or 72 . . 3 ((ab ) ∪ (ab)) = (((a ∪ (ab)) ∩ (ab )) ∪ ((a ∪ (ab)) ∩ a))
3210, 11, 313tr 65 . 2 (ab) = (((a ∪ (ab)) ∩ (ab )) ∪ ((a ∪ (ab)) ∩ a))
33 df-i1 44 . . . 4 (a1 (ab)) = (a ∪ (a ∩ (ab)))
34 lear 161 . . . . . 6 (a ∩ (ab)) ≤ (ab)
35 leid 148 . . . . . . 7 (ab) ≤ (ab)
3620, 35ler2an 173 . . . . . 6 (ab) ≤ (a ∩ (ab))
3734, 36lebi 145 . . . . 5 (a ∩ (ab)) = (ab)
3837lor 70 . . . 4 (a ∪ (a ∩ (ab))) = (a ∪ (ab))
3933, 38ax-r2 36 . . 3 (a1 (ab)) = (a ∪ (ab))
40 df-i1 44 . . . 4 ((ab) →1 a) = ((ab) ∪ ((ab) ∩ a))
41 anor3 90 . . . . . 6 (ab ) = (ab)
4241ax-r1 35 . . . . 5 (ab) = (ab )
43 lear 161 . . . . . 6 ((ab) ∩ a) ≤ a
44 leo 158 . . . . . . 7 a ≤ (ab)
45 leid 148 . . . . . . 7 aa
4644, 45ler2an 173 . . . . . 6 a ≤ ((ab) ∩ a)
4743, 46lebi 145 . . . . 5 ((ab) ∩ a) = a
4842, 472or 72 . . . 4 ((ab) ∪ ((ab) ∩ a)) = ((ab ) ∪ a)
4940, 48ax-r2 36 . . 3 ((ab) →1 a) = ((ab ) ∪ a)
5039, 492an 79 . 2 ((a1 (ab)) ∩ ((ab) →1 a)) = ((a ∪ (ab)) ∩ ((ab ) ∪ a))
519, 32, 503tr1 63 1 (ab) = ((a1 (ab)) ∩ ((ab) →1 a))
Colors of variables: term
Syntax hints:   = wb 1   wn 4  tb 5  wo 6  wa 7  0wf 9  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:  mlaconj  845
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