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Theorem oal42 935
 Description: Derivation of Godowski/Greechie Eq. II from Eq. IV.
Hypothesis
Ref Expression
oal42.1 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ ((b ∩ (a2 b)) ∪ (c ∩ (a2 c)))
Assertion
Ref Expression
oal42 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ a

Proof of Theorem oal42
StepHypRef Expression
1 oal42.1 . . 3 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ ((b ∩ (a2 b)) ∪ (c ∩ (a2 c)))
2 ancom 74 . . . . 5 (b ∩ (a2 b)) = ((a2 b) ∩ b )
3 u2lemanb 616 . . . . 5 ((a2 b) ∩ b ) = (ab )
42, 3ax-r2 36 . . . 4 (b ∩ (a2 b)) = (ab )
5 ancom 74 . . . . 5 (c ∩ (a2 c)) = ((a2 c) ∩ c )
6 u2lemanb 616 . . . . 5 ((a2 c) ∩ c ) = (ac )
75, 6ax-r2 36 . . . 4 (c ∩ (a2 c)) = (ac )
84, 72or 72 . . 3 ((b ∩ (a2 b)) ∪ (c ∩ (a2 c))) = ((ab ) ∪ (ac ))
91, 8lbtr 139 . 2 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ ((ab ) ∪ (ac ))
10 lea 160 . . 3 (ab ) ≤ a
11 lea 160 . . 3 (ac ) ≤ a
1210, 11lel2or 170 . 2 ((ab ) ∪ (ac )) ≤ a
139, 12letr 137 1 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ a
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →2 wi2 13 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-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  oa43v  1028  oa63v  1032
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