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Theorem sklem 230
 Description: Soundness lemma.
Hypothesis
Ref Expression
sklem.1 ab
Assertion
Ref Expression
sklem (ab) = 1

Proof of Theorem sklem
StepHypRef Expression
1 or12 80 . . 3 (a ∪ (ab)) = (a ∪ (ab))
2 df-t 41 . . . . . 6 1 = (aa )
32ax-r5 38 . . . . 5 (1 ∪ b) = ((aa ) ∪ b)
43ax-r1 35 . . . 4 ((aa ) ∪ b) = (1 ∪ b)
5 ax-a3 32 . . . 4 ((aa ) ∪ b) = (a ∪ (ab))
6 ax-a2 31 . . . 4 (1 ∪ b) = (b ∪ 1)
74, 5, 63tr2 64 . . 3 (a ∪ (ab)) = (b ∪ 1)
81, 7ax-r2 36 . 2 (a ∪ (ab)) = (b ∪ 1)
9 sklem.1 . . . 4 ab
109df-le2 131 . . 3 (ab) = b
1110lor 70 . 2 (a ∪ (ab)) = (ab)
12 or1 104 . 2 (b ∪ 1) = 1
138, 11, 123tr2 64 1 (ab) = 1
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6  1wt 8 This theorem was proved from axioms:  ax-a2 31  ax-a3 32  ax-a4 33  ax-r1 35  ax-r2 36  ax-r5 38 This theorem depends on definitions:  df-t 41  df-le2 131 This theorem is referenced by:  ska13  241  ska15  244  lei3  246  oaidlem1  294  id5id0  352  u1lemle1  710  u2lemle1  711  u3lemle1  712  u4lemle1  713  u5lemle1  714  lem3.3.3  1052  lem3.3.7i4e1  1069  lem3.3.7i4e2  1070  lem4.6.7  1101
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