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Theorem ka4lem 229
 Description: Lemma for KA4 soundness (AND version) - uses OL only.
Assertion
Ref Expression
ka4lem ((ab) ∪ ((ac) ≡ (bc))) = 1

Proof of Theorem ka4lem
StepHypRef Expression
1 df-a 40 . . . 4 (ab) = (ab )
21con2 67 . . 3 (ab) = (ab )
3 df-a 40 . . . . 5 (ac) = (ac )
4 df-a 40 . . . . 5 (bc) = (bc )
53, 42bi 99 . . . 4 ((ac) ≡ (bc)) = ((ac ) ≡ (bc ) )
6 conb 122 . . . . 5 ((ac ) ≡ (bc )) = ((ac ) ≡ (bc ) )
76ax-r1 35 . . . 4 ((ac ) ≡ (bc ) ) = ((ac ) ≡ (bc ))
85, 7ax-r2 36 . . 3 ((ac) ≡ (bc)) = ((ac ) ≡ (bc ))
92, 82or 72 . 2 ((ab) ∪ ((ac) ≡ (bc))) = ((ab ) ∪ ((ac ) ≡ (bc )))
10 ka4lemo 228 . 2 ((ab ) ∪ ((ac ) ≡ (bc ))) = 1
119, 10ax-r2 36 1 ((ab) ∪ ((ac) ≡ (bc))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8 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 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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