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Theorem wbctr 410
 Description: Transitive inference.
Hypotheses
Ref Expression
wbctr.1 (ab) = 1
wbctr.2 C (b, c) = 1
Assertion
Ref Expression
wbctr C (a, c) = 1

Proof of Theorem wbctr
StepHypRef Expression
1 wbctr.2 . . . 4 C (b, c) = 1
21wdf-c2 384 . . 3 (b ≡ ((bc) ∪ (bc ))) = 1
3 wbctr.1 . . 3 (ab) = 1
43wran 369 . . . 4 ((ac) ≡ (bc)) = 1
53wran 369 . . . 4 ((ac ) ≡ (bc )) = 1
64, 5w2or 372 . . 3 (((ac) ∪ (ac )) ≡ ((bc) ∪ (bc ))) = 1
72, 3, 6w3tr1 374 . 2 (a ≡ ((ac) ∪ (ac ))) = 1
87wdf-c1 383 1 C (a, c) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   C wcmtr 29 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-wom 361 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134 This theorem is referenced by:  woml7  437
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