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Theorem 3vth5 808
 Description: A 3-variable theorem.
Assertion
Ref Expression
3vth5 ((a2 b)2 (bc)) = (c ∪ ((a2 b) ∩ (c2 b)))

Proof of Theorem 3vth5
StepHypRef Expression
1 ax-a3 32 . . 3 ((bc) ∪ ((b ∪ (ab )) ∩ (bc) )) = (b ∪ (c ∪ ((b ∪ (ab )) ∩ (bc) )))
2 or12 80 . . . 4 (b ∪ (c ∪ ((b ∪ (ab )) ∩ (bc) ))) = (c ∪ (b ∪ ((b ∪ (ab )) ∩ (bc) )))
3 comorr 184 . . . . . . 7 b C (b ∪ (ab ))
4 comorr 184 . . . . . . . 8 b C (bc)
54comcom2 183 . . . . . . 7 b C (bc)
63, 5fh3 471 . . . . . 6 (b ∪ ((b ∪ (ab )) ∩ (bc) )) = ((b ∪ (b ∪ (ab ))) ∩ (b ∪ (bc) ))
7 ax-a3 32 . . . . . . . . 9 ((bb) ∪ (ab )) = (b ∪ (b ∪ (ab )))
87ax-r1 35 . . . . . . . 8 (b ∪ (b ∪ (ab ))) = ((bb) ∪ (ab ))
9 oridm 110 . . . . . . . . 9 (bb) = b
109ax-r5 38 . . . . . . . 8 ((bb) ∪ (ab )) = (b ∪ (ab ))
118, 10ax-r2 36 . . . . . . 7 (b ∪ (b ∪ (ab ))) = (b ∪ (ab ))
12 ancom 74 . . . . . . . . . 10 (cb ) = (bc )
13 anor3 90 . . . . . . . . . 10 (bc ) = (bc)
1412, 13ax-r2 36 . . . . . . . . 9 (cb ) = (bc)
1514ax-r1 35 . . . . . . . 8 (bc) = (cb )
1615lor 70 . . . . . . 7 (b ∪ (bc) ) = (b ∪ (cb ))
1711, 162an 79 . . . . . 6 ((b ∪ (b ∪ (ab ))) ∩ (b ∪ (bc) )) = ((b ∪ (ab )) ∩ (b ∪ (cb )))
186, 17ax-r2 36 . . . . 5 (b ∪ ((b ∪ (ab )) ∩ (bc) )) = ((b ∪ (ab )) ∩ (b ∪ (cb )))
1918lor 70 . . . 4 (c ∪ (b ∪ ((b ∪ (ab )) ∩ (bc) ))) = (c ∪ ((b ∪ (ab )) ∩ (b ∪ (cb ))))
202, 19ax-r2 36 . . 3 (b ∪ (c ∪ ((b ∪ (ab )) ∩ (bc) ))) = (c ∪ ((b ∪ (ab )) ∩ (b ∪ (cb ))))
211, 20ax-r2 36 . 2 ((bc) ∪ ((b ∪ (ab )) ∩ (bc) )) = (c ∪ ((b ∪ (ab )) ∩ (b ∪ (cb ))))
22 df-i2 45 . . 3 ((a2 b)2 (bc)) = ((bc) ∪ ((a2 b) ∩ (bc) ))
23 df-i2 45 . . . . . . . 8 (a2 b) = (b ∪ (ab ))
2423ax-r1 35 . . . . . . 7 (b ∪ (ab )) = (a2 b)
25 ax-a1 30 . . . . . . 7 (a2 b) = (a2 b)
2624, 25ax-r2 36 . . . . . 6 (b ∪ (ab )) = (a2 b)
2726ran 78 . . . . 5 ((b ∪ (ab )) ∩ (bc) ) = ((a2 b) ∩ (bc) )
2827lor 70 . . . 4 ((bc) ∪ ((b ∪ (ab )) ∩ (bc) )) = ((bc) ∪ ((a2 b) ∩ (bc) ))
2928ax-r1 35 . . 3 ((bc) ∪ ((a2 b) ∩ (bc) )) = ((bc) ∪ ((b ∪ (ab )) ∩ (bc) ))
3022, 29ax-r2 36 . 2 ((a2 b)2 (bc)) = ((bc) ∪ ((b ∪ (ab )) ∩ (bc) ))
31 df-i2 45 . . . 4 (c2 b) = (b ∪ (cb ))
3223, 312an 79 . . 3 ((a2 b) ∩ (c2 b)) = ((b ∪ (ab )) ∩ (b ∪ (cb )))
3332lor 70 . 2 (c ∪ ((a2 b) ∩ (c2 b))) = (c ∪ ((b ∪ (ab )) ∩ (b ∪ (cb ))))
3421, 30, 333tr1 63 1 ((a2 b)2 (bc)) = (c ∪ ((a2 b) ∩ (c2 b)))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →2 wi2 13 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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  3vth6  809  3vth7  810
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