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Theorem ud1lem3 562
Description: Lemma for unified disjunction. (Contributed by NM, 23-Nov-1997.)
Assertion
Ref Expression
ud1lem3 ((a1 b) →1 (ab)) = (ab)

Proof of Theorem ud1lem3
StepHypRef Expression
1 df-i1 44 . 2 ((a1 b) →1 (ab)) = ((a1 b) ∪ ((a1 b) ∩ (ab)))
2 ud1lem0c 277 . . . 4 (a1 b) = (a ∩ (ab ))
32con3 68 . . . . 5 (a1 b) = (a ∩ (ab ))
43ran 78 . . . 4 ((a1 b) ∩ (ab)) = ((a ∩ (ab )) ∩ (ab))
52, 42or 72 . . 3 ((a1 b) ∪ ((a1 b) ∩ (ab))) = ((a ∩ (ab )) ∪ ((a ∩ (ab )) ∩ (ab)))
6 comid 187 . . . . . 6 (a ∩ (ab )) C (a ∩ (ab ))
76comcom2 183 . . . . 5 (a ∩ (ab )) C (a ∩ (ab ))
8 comor1 461 . . . . . . 7 (ab) C a
98comcom2 183 . . . . . . . 8 (ab) C a
10 comor2 462 . . . . . . . . 9 (ab) C b
1110comcom2 183 . . . . . . . 8 (ab) C b
129, 11com2or 483 . . . . . . 7 (ab) C (ab )
138, 12com2an 484 . . . . . 6 (ab) C (a ∩ (ab ))
1413comcom 453 . . . . 5 (a ∩ (ab )) C (ab)
157, 14fh3 471 . . . 4 ((a ∩ (ab )) ∪ ((a ∩ (ab )) ∩ (ab))) = (((a ∩ (ab )) ∪ (a ∩ (ab )) ) ∩ ((a ∩ (ab )) ∪ (ab)))
16 ancom 74 . . . . 5 (((a ∩ (ab )) ∪ (a ∩ (ab )) ) ∩ ((a ∩ (ab )) ∪ (ab))) = (((a ∩ (ab )) ∪ (ab)) ∩ ((a ∩ (ab )) ∪ (a ∩ (ab )) ))
17 df-t 41 . . . . . . . 8 1 = ((a ∩ (ab )) ∪ (a ∩ (ab )) )
1817ax-r1 35 . . . . . . 7 ((a ∩ (ab )) ∪ (a ∩ (ab )) ) = 1
1918lan 77 . . . . . 6 (((a ∩ (ab )) ∪ (ab)) ∩ ((a ∩ (ab )) ∪ (a ∩ (ab )) )) = (((a ∩ (ab )) ∪ (ab)) ∩ 1)
20 an1 106 . . . . . . 7 (((a ∩ (ab )) ∪ (ab)) ∩ 1) = ((a ∩ (ab )) ∪ (ab))
21 comorr 184 . . . . . . . . 9 a C (ab)
22 comorr 184 . . . . . . . . . . 11 a C (ab )
2322comcom2 183 . . . . . . . . . 10 a C (ab )
2423comcom5 458 . . . . . . . . 9 a C (ab )
2521, 24fh4r 476 . . . . . . . 8 ((a ∩ (ab )) ∪ (ab)) = ((a ∪ (ab)) ∩ ((ab ) ∪ (ab)))
26 ax-a2 31 . . . . . . . . . . 11 ((ab ) ∪ (ab)) = ((ab) ∪ (ab ))
27 or4 84 . . . . . . . . . . . 12 ((ab) ∪ (ab )) = ((aa ) ∪ (bb ))
28 df-t 41 . . . . . . . . . . . . . . 15 1 = (bb )
2928ax-r1 35 . . . . . . . . . . . . . 14 (bb ) = 1
3029lor 70 . . . . . . . . . . . . 13 ((aa ) ∪ (bb )) = ((aa ) ∪ 1)
31 or1 104 . . . . . . . . . . . . 13 ((aa ) ∪ 1) = 1
3230, 31ax-r2 36 . . . . . . . . . . . 12 ((aa ) ∪ (bb )) = 1
3327, 32ax-r2 36 . . . . . . . . . . 11 ((ab) ∪ (ab )) = 1
3426, 33ax-r2 36 . . . . . . . . . 10 ((ab ) ∪ (ab)) = 1
3534lan 77 . . . . . . . . 9 ((a ∪ (ab)) ∩ ((ab ) ∪ (ab))) = ((a ∪ (ab)) ∩ 1)
36 an1 106 . . . . . . . . . 10 ((a ∪ (ab)) ∩ 1) = (a ∪ (ab))
37 ax-a3 32 . . . . . . . . . . . 12 ((aa) ∪ b) = (a ∪ (ab))
3837ax-r1 35 . . . . . . . . . . 11 (a ∪ (ab)) = ((aa) ∪ b)
39 oridm 110 . . . . . . . . . . . 12 (aa) = a
4039ax-r5 38 . . . . . . . . . . 11 ((aa) ∪ b) = (ab)
4138, 40ax-r2 36 . . . . . . . . . 10 (a ∪ (ab)) = (ab)
4236, 41ax-r2 36 . . . . . . . . 9 ((a ∪ (ab)) ∩ 1) = (ab)
4335, 42ax-r2 36 . . . . . . . 8 ((a ∪ (ab)) ∩ ((ab ) ∪ (ab))) = (ab)
4425, 43ax-r2 36 . . . . . . 7 ((a ∩ (ab )) ∪ (ab)) = (ab)
4520, 44ax-r2 36 . . . . . 6 (((a ∩ (ab )) ∪ (ab)) ∩ 1) = (ab)
4619, 45ax-r2 36 . . . . 5 (((a ∩ (ab )) ∪ (ab)) ∩ ((a ∩ (ab )) ∪ (a ∩ (ab )) )) = (ab)
4716, 46ax-r2 36 . . . 4 (((a ∩ (ab )) ∪ (a ∩ (ab )) ) ∩ ((a ∩ (ab )) ∪ (ab))) = (ab)
4815, 47ax-r2 36 . . 3 ((a ∩ (ab )) ∪ ((a ∩ (ab )) ∩ (ab))) = (ab)
495, 48ax-r2 36 . 2 ((a1 b) ∪ ((a1 b) ∩ (ab))) = (ab)
501, 49ax-r2 36 1 ((a1 b) →1 (ab)) = (ab)
Colors of variables: term
Syntax hints:   = wb 1   wn 4  wo 6  wa 7  1wt 8  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:  ud1  595
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