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Theorem ud4lem3 585
Description: Lemma for unified disjunction. (Contributed by NM, 23-Nov-1997.)
Assertion
Ref Expression
ud4lem3 ((a4 b) →4 (ab)) = (ab)

Proof of Theorem ud4lem3
StepHypRef Expression
1 df-i4 47 . 2 ((a4 b) →4 (ab)) = ((((a4 b) ∩ (ab)) ∪ ((a4 b) ∩ (ab))) ∪ (((a4 b) ∪ (ab)) ∩ (ab) ))
2 ud4lem3a 583 . . . . . 6 ((a4 b) ∩ (ab)) = (a4 b)
32lor 70 . . . . 5 (((a4 b) ∩ (ab)) ∪ ((a4 b) ∩ (ab))) = (((a4 b) ∩ (ab)) ∪ (a4 b) )
4 comid 187 . . . . . . . 8 (a4 b) C (a4 b)
54comcom2 183 . . . . . . 7 (a4 b) C (a4 b)
6 df-i4 47 . . . . . . . 8 (a4 b) = (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))
7 comor1 461 . . . . . . . . . . . 12 (ab) C a
8 comor2 462 . . . . . . . . . . . 12 (ab) C b
97, 8com2an 484 . . . . . . . . . . 11 (ab) C (ab)
107comcom2 183 . . . . . . . . . . . 12 (ab) C a
1110, 8com2an 484 . . . . . . . . . . 11 (ab) C (ab)
129, 11com2or 483 . . . . . . . . . 10 (ab) C ((ab) ∪ (ab))
1310, 8com2or 483 . . . . . . . . . . 11 (ab) C (ab)
148comcom2 183 . . . . . . . . . . 11 (ab) C b
1513, 14com2an 484 . . . . . . . . . 10 (ab) C ((ab) ∩ b )
1612, 15com2or 483 . . . . . . . . 9 (ab) C (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))
1716comcom 453 . . . . . . . 8 (((ab) ∪ (ab)) ∪ ((ab) ∩ b )) C (ab)
186, 17bctr 181 . . . . . . 7 (a4 b) C (ab)
195, 18fh4r 476 . . . . . 6 (((a4 b) ∩ (ab)) ∪ (a4 b) ) = (((a4 b) ∪ (a4 b) ) ∩ ((ab) ∪ (a4 b) ))
20 ancom 74 . . . . . . 7 (((a4 b) ∪ (a4 b) ) ∩ ((ab) ∪ (a4 b) )) = (((ab) ∪ (a4 b) ) ∩ ((a4 b) ∪ (a4 b) ))
21 ax-a2 31 . . . . . . . . . 10 ((ab) ∪ (a4 b) ) = ((a4 b) ∪ (ab))
22 ud4lem3b 584 . . . . . . . . . 10 ((a4 b) ∪ (ab)) = (ab)
2321, 22ax-r2 36 . . . . . . . . 9 ((ab) ∪ (a4 b) ) = (ab)
24 df-t 41 . . . . . . . . . 10 1 = ((a4 b) ∪ (a4 b) )
2524ax-r1 35 . . . . . . . . 9 ((a4 b) ∪ (a4 b) ) = 1
2623, 252an 79 . . . . . . . 8 (((ab) ∪ (a4 b) ) ∩ ((a4 b) ∪ (a4 b) )) = ((ab) ∩ 1)
27 an1 106 . . . . . . . 8 ((ab) ∩ 1) = (ab)
2826, 27ax-r2 36 . . . . . . 7 (((ab) ∪ (a4 b) ) ∩ ((a4 b) ∪ (a4 b) )) = (ab)
2920, 28ax-r2 36 . . . . . 6 (((a4 b) ∪ (a4 b) ) ∩ ((ab) ∪ (a4 b) )) = (ab)
3019, 29ax-r2 36 . . . . 5 (((a4 b) ∩ (ab)) ∪ (a4 b) ) = (ab)
313, 30ax-r2 36 . . . 4 (((a4 b) ∩ (ab)) ∪ ((a4 b) ∩ (ab))) = (ab)
3222ran 78 . . . . 5 (((a4 b) ∪ (ab)) ∩ (ab) ) = ((ab) ∩ (ab) )
33 dff 101 . . . . . 6 0 = ((ab) ∩ (ab) )
3433ax-r1 35 . . . . 5 ((ab) ∩ (ab) ) = 0
3532, 34ax-r2 36 . . . 4 (((a4 b) ∪ (ab)) ∩ (ab) ) = 0
3631, 352or 72 . . 3 ((((a4 b) ∩ (ab)) ∪ ((a4 b) ∩ (ab))) ∪ (((a4 b) ∪ (ab)) ∩ (ab) )) = ((ab) ∪ 0)
37 or0 102 . . 3 ((ab) ∪ 0) = (ab)
3836, 37ax-r2 36 . 2 ((((a4 b) ∩ (ab)) ∪ ((a4 b) ∩ (ab))) ∪ (((a4 b) ∪ (ab)) ∩ (ab) )) = (ab)
391, 38ax-r2 36 1 ((a4 b) →4 (ab)) = (ab)
Colors of variables: term
Syntax hints:   = wb 1   wn 4  wo 6  wa 7  1wt 8  0wf 9  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:  ud4  598
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