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Theorem u5lemc2 690
 Description: Commutation theorem for relevance implication. (Contributed by NM, 14-Dec-1997.)
Hypotheses
Ref Expression
ulemc2.1 a C b
ulemc2.2 a C c
Assertion
Ref Expression
u5lemc2 a C (b5 c)

Proof of Theorem u5lemc2
StepHypRef Expression
1 ulemc2.1 . . . . 5 a C b
2 ulemc2.2 . . . . 5 a C c
31, 2com2an 484 . . . 4 a C (bc)
41comcom2 183 . . . . 5 a C b
54, 2com2an 484 . . . 4 a C (bc)
63, 5com2or 483 . . 3 a C ((bc) ∪ (bc))
72comcom2 183 . . . 4 a C c
84, 7com2an 484 . . 3 a C (bc )
96, 8com2or 483 . 2 a C (((bc) ∪ (bc)) ∪ (bc ))
10 df-i5 48 . . 3 (b5 c) = (((bc) ∪ (bc)) ∪ (bc ))
1110ax-r1 35 . 2 (((bc) ∪ (bc)) ∪ (bc )) = (b5 c)
129, 11cbtr 182 1 a C (b5 c)
 Colors of variables: term Syntax hints:   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →5 wi5 16 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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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