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Theorem gomaex3lem4 917
Description: Lemma for Godowski 6-var -> Mayet Example 3. (Contributed by NM, 29-Nov-1999.)
Hypothesis
Ref Expression
gomaex3lem4.9 p = ((ab) →1 (de) )
Assertion
Ref Expression
gomaex3lem4 ((ab) ∩ (de) ) ≤ p

Proof of Theorem gomaex3lem4
StepHypRef Expression
1 leor 159 . 2 ((ab) ∩ (de) ) ≤ ((ab) ∪ ((ab) ∩ (de) ))
2 ax-a1 30 . . 3 ((ab) →1 (de) ) = ((ab) →1 (de) )
3 df-i1 44 . . . 4 ((ab) →1 (de) ) = ((ab) ∪ ((ab) ∩ (de) ))
43ax-r1 35 . . 3 ((ab) ∪ ((ab) ∩ (de) )) = ((ab) →1 (de) )
5 gomaex3lem4.9 . . . 4 p = ((ab) →1 (de) )
65ax-r4 37 . . 3 p = ((ab) →1 (de) )
72, 4, 63tr1 63 . 2 ((ab) ∪ ((ab) ∩ (de) )) = p
81, 7lbtr 139 1 ((ab) ∩ (de) ) ≤ p
Colors of variables: term
Syntax hints:   = wb 1  wle 2   wn 4  wo 6  wa 7  1 wi1 12
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-i1 44  df-le1 130  df-le2 131
This theorem is referenced by:  gomaex3lem9  922
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