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Theorem 3vth7 810
 Description: A 3-variable theorem.
Assertion
Ref Expression
3vth7 ((a2 b)2 (bc)) = (a2 (bc))

Proof of Theorem 3vth7
StepHypRef Expression
1 df-i2 45 . . . . 5 (a2 b) = (b ∪ (ab ))
2 df-i2 45 . . . . 5 (c2 b) = (b ∪ (cb ))
31, 22an 79 . . . 4 ((a2 b) ∩ (c2 b)) = ((b ∪ (ab )) ∩ (b ∪ (cb )))
4 anass 76 . . . . . . . . . 10 ((ab ) ∩ c ) = (a ∩ (bc ))
54ax-r1 35 . . . . . . . . 9 (a ∩ (bc )) = ((ab ) ∩ c )
6 anor3 90 . . . . . . . . . 10 (bc ) = (bc)
76lan 77 . . . . . . . . 9 (a ∩ (bc )) = (a ∩ (bc) )
8 an32 83 . . . . . . . . 9 ((ab ) ∩ c ) = ((ac ) ∩ b )
95, 7, 83tr2 64 . . . . . . . 8 (a ∩ (bc) ) = ((ac ) ∩ b )
10 anidm 111 . . . . . . . . . 10 (bb ) = b
1110lan 77 . . . . . . . . 9 ((ac ) ∩ (bb )) = ((ac ) ∩ b )
1211ax-r1 35 . . . . . . . 8 ((ac ) ∩ b ) = ((ac ) ∩ (bb ))
13 an4 86 . . . . . . . 8 ((ac ) ∩ (bb )) = ((ab ) ∩ (cb ))
149, 12, 133tr 65 . . . . . . 7 (a ∩ (bc) ) = ((ab ) ∩ (cb ))
1514lor 70 . . . . . 6 (b ∪ (a ∩ (bc) )) = (b ∪ ((ab ) ∩ (cb )))
16 comanr2 465 . . . . . . . 8 b C (ab )
1716comcom6 459 . . . . . . 7 b C (ab )
18 comanr2 465 . . . . . . . 8 b C (cb )
1918comcom6 459 . . . . . . 7 b C (cb )
2017, 19fh3 471 . . . . . 6 (b ∪ ((ab ) ∩ (cb ))) = ((b ∪ (ab )) ∩ (b ∪ (cb )))
2115, 20ax-r2 36 . . . . 5 (b ∪ (a ∩ (bc) )) = ((b ∪ (ab )) ∩ (b ∪ (cb )))
2221ax-r1 35 . . . 4 ((b ∪ (ab )) ∩ (b ∪ (cb ))) = (b ∪ (a ∩ (bc) ))
233, 22ax-r2 36 . . 3 ((a2 b) ∩ (c2 b)) = (b ∪ (a ∩ (bc) ))
2423lor 70 . 2 (c ∪ ((a2 b) ∩ (c2 b))) = (c ∪ (b ∪ (a ∩ (bc) )))
25 3vth5 808 . 2 ((a2 b)2 (bc)) = (c ∪ ((a2 b) ∩ (c2 b)))
26 df-i2 45 . . 3 (a2 (bc)) = ((bc) ∪ (a ∩ (bc) ))
27 ax-a3 32 . . 3 ((bc) ∪ (a ∩ (bc) )) = (b ∪ (c ∪ (a ∩ (bc) )))
28 or12 80 . . 3 (b ∪ (c ∪ (a ∩ (bc) ))) = (c ∪ (b ∪ (a ∩ (bc) )))
2926, 27, 283tr 65 . 2 (a2 (bc)) = (c ∪ (b ∪ (a ∩ (bc) )))
3024, 25, 293tr1 63 1 ((a2 b)2 (bc)) = (a2 (bc))
 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:  3vth8  811
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