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Theorem 3vcom 813
 Description: 3-variable commutation theorem.
Assertion
Ref Expression
3vcom ((a1 c) ∪ (b1 c)) C ((a1 c) ∩ (b1 c))

Proof of Theorem 3vcom
StepHypRef Expression
1 oran3 93 . . . . 5 ((a1 c) ∪ (b1 c) ) = ((a1 c) ∩ (b1 c))
21ax-r1 35 . . . 4 ((a1 c) ∩ (b1 c)) = ((a1 c) ∪ (b1 c) )
3 u1lem9ab 779 . . . . . 6 (a1 c) ≤ (a1 c)
4 u1lem9ab 779 . . . . . 6 (b1 c) ≤ (b1 c)
53, 4le2or 168 . . . . 5 ((a1 c) ∪ (b1 c) ) ≤ ((a1 c) ∪ (b1 c))
65lecom 180 . . . 4 ((a1 c) ∪ (b1 c) ) C ((a1 c) ∪ (b1 c))
72, 6bctr 181 . . 3 ((a1 c) ∩ (b1 c)) C ((a1 c) ∪ (b1 c))
87comcom6 459 . 2 ((a1 c) ∩ (b1 c)) C ((a1 c) ∪ (b1 c))
98comcom 453 1 ((a1 c) ∪ (b1 c)) C ((a1 c) ∩ (b1 c))
 Colors of variables: term Syntax hints:   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12 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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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