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Theorem 3vth4 807
 Description: A 3-variable theorem.
Assertion
Ref Expression
3vth4 ((a2 b)2 (bc)) = ((a2 c)2 (bc))

Proof of Theorem 3vth4
StepHypRef Expression
1 3vth2 805 . . . 4 ((a2 b) ∩ (bc) ) = ((a2 c) ∩ (bc) )
2 ax-a1 30 . . . . 5 (a2 b) = (a2 b)
32ran 78 . . . 4 ((a2 b) ∩ (bc) ) = ((a2 b) ∩ (bc) )
4 ax-a1 30 . . . . 5 (a2 c) = (a2 c)
54ran 78 . . . 4 ((a2 c) ∩ (bc) ) = ((a2 c) ∩ (bc) )
61, 3, 53tr2 64 . . 3 ((a2 b) ∩ (bc) ) = ((a2 c) ∩ (bc) )
76lor 70 . 2 ((bc) ∪ ((a2 b) ∩ (bc) )) = ((bc) ∪ ((a2 c) ∩ (bc) ))
8 df-i2 45 . 2 ((a2 b)2 (bc)) = ((bc) ∪ ((a2 b) ∩ (bc) ))
9 df-i2 45 . 2 ((a2 c)2 (bc)) = ((bc) ∪ ((a2 c) ∩ (bc) ))
107, 8, 93tr1 63 1 ((a2 b)2 (bc)) = ((a2 c)2 (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-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 This theorem is referenced by:  3vth6  809
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