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Theorem ud4lem3b 584
Description: Lemma for unified disjunction. (Contributed by NM, 23-Nov-1997.)
Assertion
Ref Expression
ud4lem3b ((a4 b) ∪ (ab)) = (ab)

Proof of Theorem ud4lem3b
StepHypRef Expression
1 ud4lem0c 280 . . 3 (a4 b) = (((ab ) ∩ (ab )) ∩ ((ab ) ∪ b))
21ax-r5 38 . 2 ((a4 b) ∪ (ab)) = ((((ab ) ∩ (ab )) ∩ ((ab ) ∪ b)) ∪ (ab))
3 comor1 461 . . . . . . 7 (ab) C a
43comcom2 183 . . . . . 6 (ab) C a
5 comor2 462 . . . . . . 7 (ab) C b
65comcom2 183 . . . . . 6 (ab) C b
74, 6com2or 483 . . . . 5 (ab) C (ab )
83, 6com2or 483 . . . . 5 (ab) C (ab )
97, 8com2an 484 . . . 4 (ab) C ((ab ) ∩ (ab ))
103, 6com2an 484 . . . . 5 (ab) C (ab )
1110, 5com2or 483 . . . 4 (ab) C ((ab ) ∪ b)
129, 11fh3r 475 . . 3 ((((ab ) ∩ (ab )) ∩ ((ab ) ∪ b)) ∪ (ab)) = ((((ab ) ∩ (ab )) ∪ (ab)) ∩ (((ab ) ∪ b) ∪ (ab)))
137, 8fh3r 475 . . . . . . 7 (((ab ) ∩ (ab )) ∪ (ab)) = (((ab ) ∪ (ab)) ∩ ((ab ) ∪ (ab)))
14 ax-a2 31 . . . . . . . . 9 ((ab ) ∪ (ab)) = ((ab) ∪ (ab ))
15 or4 84 . . . . . . . . . 10 ((ab) ∪ (ab )) = ((aa ) ∪ (bb ))
16 df-t 41 . . . . . . . . . . . . 13 1 = (bb )
1716lor 70 . . . . . . . . . . . 12 ((aa ) ∪ 1) = ((aa ) ∪ (bb ))
1817ax-r1 35 . . . . . . . . . . 11 ((aa ) ∪ (bb )) = ((aa ) ∪ 1)
19 or1 104 . . . . . . . . . . 11 ((aa ) ∪ 1) = 1
2018, 19ax-r2 36 . . . . . . . . . 10 ((aa ) ∪ (bb )) = 1
2115, 20ax-r2 36 . . . . . . . . 9 ((ab) ∪ (ab )) = 1
2214, 21ax-r2 36 . . . . . . . 8 ((ab ) ∪ (ab)) = 1
23 ax-a2 31 . . . . . . . . 9 ((ab ) ∪ (ab)) = ((ab) ∪ (ab ))
24 or4 84 . . . . . . . . . 10 ((ab) ∪ (ab )) = ((aa) ∪ (bb ))
2516lor 70 . . . . . . . . . . . 12 ((aa) ∪ 1) = ((aa) ∪ (bb ))
2625ax-r1 35 . . . . . . . . . . 11 ((aa) ∪ (bb )) = ((aa) ∪ 1)
27 or1 104 . . . . . . . . . . 11 ((aa) ∪ 1) = 1
2826, 27ax-r2 36 . . . . . . . . . 10 ((aa) ∪ (bb )) = 1
2924, 28ax-r2 36 . . . . . . . . 9 ((ab) ∪ (ab )) = 1
3023, 29ax-r2 36 . . . . . . . 8 ((ab ) ∪ (ab)) = 1
3122, 302an 79 . . . . . . 7 (((ab ) ∪ (ab)) ∩ ((ab ) ∪ (ab))) = (1 ∩ 1)
3213, 31ax-r2 36 . . . . . 6 (((ab ) ∩ (ab )) ∪ (ab)) = (1 ∩ 1)
33 an1 106 . . . . . 6 (1 ∩ 1) = 1
3432, 33ax-r2 36 . . . . 5 (((ab ) ∩ (ab )) ∪ (ab)) = 1
35 lea 160 . . . . . . 7 (ab ) ≤ a
3635leror 152 . . . . . 6 ((ab ) ∪ b) ≤ (ab)
3736df-le2 131 . . . . 5 (((ab ) ∪ b) ∪ (ab)) = (ab)
3834, 372an 79 . . . 4 ((((ab ) ∩ (ab )) ∪ (ab)) ∩ (((ab ) ∪ b) ∪ (ab))) = (1 ∩ (ab))
39 ancom 74 . . . . 5 (1 ∩ (ab)) = ((ab) ∩ 1)
40 an1 106 . . . . 5 ((ab) ∩ 1) = (ab)
4139, 40ax-r2 36 . . . 4 (1 ∩ (ab)) = (ab)
4238, 41ax-r2 36 . . 3 ((((ab ) ∩ (ab )) ∪ (ab)) ∩ (((ab ) ∪ b) ∪ (ab))) = (ab)
4312, 42ax-r2 36 . 2 ((((ab ) ∩ (ab )) ∩ ((ab ) ∪ b)) ∪ (ab)) = (ab)
442, 43ax-r2 36 1 ((a4 b) ∪ (ab)) = (ab)
Colors of variables: term
Syntax hints:   = wb 1   wn 4  wo 6  wa 7  1wt 8  4 wi4 15
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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  ud4lem3  585
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