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Theorem u2lemc2 687
 Description: Commutation theorem for Dishkant implication.
Hypotheses
Ref Expression
ulemc2.1 a C b
ulemc2.2 a C c
Assertion
Ref Expression
u2lemc2 a C (b2 c)

Proof of Theorem u2lemc2
StepHypRef Expression
1 ulemc2.2 . . 3 a C c
2 ulemc2.1 . . . . 5 a C b
32comcom2 183 . . . 4 a C b
41comcom2 183 . . . 4 a C c
53, 4com2an 484 . . 3 a C (bc )
61, 5com2or 483 . 2 a C (c ∪ (bc ))
7 df-i2 45 . . 3 (b2 c) = (c ∪ (bc ))
87ax-r1 35 . 2 (c ∪ (bc )) = (b2 c)
96, 8cbtr 182 1 a C (b2 c)
 Colors of variables: term Syntax hints:   C wc 3  ⊥ 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-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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u2lemc5  697
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