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Theorem comi31 508
 Description: Commutation theorem.
Assertion
Ref Expression
comi31 a C (a3 b)

Proof of Theorem comi31
StepHypRef Expression
1 coman1 185 . . . . . . 7 (ab) C a
21comcom 453 . . . . . 6 a C (ab)
32comcom2 183 . . . . 5 a C (ab)
43comcom5 458 . . . 4 a C (ab)
5 coman1 185 . . . . . . 7 (ab ) C a
65comcom 453 . . . . . 6 a C (ab )
76comcom2 183 . . . . 5 a C (ab )
87comcom5 458 . . . 4 a C (ab )
94, 8com2or 483 . . 3 a C ((ab) ∪ (ab ))
10 coman1 185 . . . 4 (a ∩ (ab)) C a
1110comcom 453 . . 3 a C (a ∩ (ab))
129, 11com2or 483 . 2 a C (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
13 df-i3 46 . . 3 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
1413ax-r1 35 . 2 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) = (a3 b)
1512, 14cbtr 182 1 a C (a3 b)
 Colors of variables: term Syntax hints:   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  i3abs3  524  u3lemc1  682  u3lemc5  698  u3lem1  736  u3lem2  746  u3lem5  763  u3lem6  767  u3lem7  774  u3lem8  783  u3lem9  784  u3lem13a  789  u3lem13b  790
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