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Theorem oa4b 959
 Description: Derivation of 4-OA law variant.
Hypothesis
Ref Expression
oa4b.1 ((a1 g) ∩ (a ∪ (c ∩ (((ac) ∪ ((a1 g) ∩ (c1 g))) ∪ (((ae) ∪ ((a1 g) ∩ (e1 g))) ∩ ((ce) ∪ ((c1 g) ∩ (e1 g)))))))) ≤ (((ag) ∪ (cg)) ∪ (eg))
Assertion
Ref Expression
oa4b ((a1 g) ∩ (a ∪ (c ∩ (((ac) ∪ ((a1 g) ∩ (c1 g))) ∪ (((ae) ∪ ((a1 g) ∩ (e1 g))) ∩ ((ce) ∪ ((c1 g) ∩ (e1 g)))))))) ≤ g

Proof of Theorem oa4b
StepHypRef Expression
1 oa4b.1 . 2 ((a1 g) ∩ (a ∪ (c ∩ (((ac) ∪ ((a1 g) ∩ (c1 g))) ∪ (((ae) ∪ ((a1 g) ∩ (e1 g))) ∩ ((ce) ∪ ((c1 g) ∩ (e1 g)))))))) ≤ (((ag) ∪ (cg)) ∪ (eg))
2 lear 161 . . . 4 (ag) ≤ g
3 lear 161 . . . 4 (cg) ≤ g
42, 3lel2or 170 . . 3 ((ag) ∪ (cg)) ≤ g
5 lear 161 . . 3 (eg) ≤ g
64, 5lel2or 170 . 2 (((ag) ∪ (cg)) ∪ (eg)) ≤ g
71, 6letr 137 1 ((a1 g) ∩ (a ∪ (c ∩ (((ac) ∪ ((a1 g) ∩ (c1 g))) ∪ (((ae) ∪ ((a1 g) ∩ (e1 g))) ∩ ((ce) ∪ ((c1 g) ∩ (e1 g)))))))) ≤ g
 Colors of variables: term Syntax hints:   ≤ wle 2   ∪ wo 6   ∩ wa 7   →1 wi1 12 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  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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