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Theorem com2i3 509
 Description: Commutation theorem.
Hypotheses
Ref Expression
com2i3.1 a C b
com2i3.2 a C c
Assertion
Ref Expression
com2i3 a C (b3 c)

Proof of Theorem com2i3
StepHypRef Expression
1 com2i3.1 . . . . . 6 a C b
21comcom2 183 . . . . 5 a C b
3 com2i3.2 . . . . 5 a C c
42, 3com2an 484 . . . 4 a C (bc)
53comcom2 183 . . . . 5 a C c
62, 5com2an 484 . . . 4 a C (bc )
74, 6com2or 483 . . 3 a C ((bc) ∪ (bc ))
82, 3com2or 483 . . . 4 a C (bc)
91, 8com2an 484 . . 3 a C (b ∩ (bc))
107, 9com2or 483 . 2 a C (((bc) ∪ (bc )) ∪ (b ∩ (bc)))
11 df-i3 46 . . 3 (b3 c) = (((bc) ∪ (bc )) ∪ (b ∩ (bc)))
1211ax-r1 35 . 2 (((bc) ∪ (bc )) ∪ (b ∩ (bc))) = (b3 c)
1310, 12cbtr 182 1 a C (b3 c)
 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:  comi32  510  u3lemc2  688
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