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Theorem elimconslem 867
 Description: Lemma for consequent elimination law. (Contributed by NM, 3-Mar-2002.)
Hypotheses
Ref Expression
elimcons.1 (a1 c) = (b1 c)
elimcons.2 (ac) ≤ (bc )
Assertion
Ref Expression
elimconslem a ≤ (bc )

Proof of Theorem elimconslem
StepHypRef Expression
1 df-t 41 . . . . . . 7 1 = ((bc ) ∪ (bc ) )
2 elimcons.2 . . . . . . . . . 10 (ac) ≤ (bc )
32lecon 154 . . . . . . . . 9 (bc ) ≤ (ac)
4 oran3 93 . . . . . . . . . 10 (ac ) = (ac)
54ax-r1 35 . . . . . . . . 9 (ac) = (ac )
63, 5lbtr 139 . . . . . . . 8 (bc ) ≤ (ac )
76lelor 166 . . . . . . 7 ((bc ) ∪ (bc ) ) ≤ ((bc ) ∪ (ac ))
81, 7bltr 138 . . . . . 6 1 ≤ ((bc ) ∪ (ac ))
98lelan 167 . . . . 5 (a ∩ 1) ≤ (a ∩ ((bc ) ∪ (ac )))
10 an1 106 . . . . 5 (a ∩ 1) = a
11 comor1 461 . . . . . . 7 (ac ) C a
1211comcom7 460 . . . . . 6 (ac ) C a
13 df-a 40 . . . . . . . . . 10 (ac) = (ac )
1413ax-r1 35 . . . . . . . . 9 (ac ) = (ac)
1514, 2bltr 138 . . . . . . . 8 (ac ) ≤ (bc )
1615lecom 180 . . . . . . 7 (ac ) C (bc )
1716comcom6 459 . . . . . 6 (ac ) C (bc )
1812, 17fh2c 477 . . . . 5 (a ∩ ((bc ) ∪ (ac ))) = ((a ∩ (bc )) ∪ (a ∩ (ac )))
199, 10, 18le3tr2 141 . . . 4 a ≤ ((a ∩ (bc )) ∪ (a ∩ (ac )))
20 elimcons.1 . . . . . . . . 9 (a1 c) = (b1 c)
21 df-i1 44 . . . . . . . . 9 (a1 c) = (a ∪ (ac))
22 df-i1 44 . . . . . . . . 9 (b1 c) = (b ∪ (bc))
2320, 21, 223tr2 64 . . . . . . . 8 (a ∪ (ac)) = (b ∪ (bc))
2413lor 70 . . . . . . . 8 (a ∪ (ac)) = (a ∪ (ac ) )
25 df-a 40 . . . . . . . . 9 (bc) = (bc )
2625lor 70 . . . . . . . 8 (b ∪ (bc)) = (b ∪ (bc ) )
2723, 24, 263tr2 64 . . . . . . 7 (a ∪ (ac ) ) = (b ∪ (bc ) )
2827ax-r4 37 . . . . . 6 (a ∪ (ac ) ) = (b ∪ (bc ) )
29 df-a 40 . . . . . 6 (a ∩ (ac )) = (a ∪ (ac ) )
30 df-a 40 . . . . . 6 (b ∩ (bc )) = (b ∪ (bc ) )
3128, 29, 303tr1 63 . . . . 5 (a ∩ (ac )) = (b ∩ (bc ))
3231lor 70 . . . 4 ((a ∩ (bc )) ∪ (a ∩ (ac ))) = ((a ∩ (bc )) ∪ (b ∩ (bc )))
3319, 32lbtr 139 . . 3 a ≤ ((a ∩ (bc )) ∪ (b ∩ (bc )))
34 lear 161 . . . 4 (a ∩ (bc )) ≤ (bc )
3534leror 152 . . 3 ((a ∩ (bc )) ∪ (b ∩ (bc ))) ≤ ((bc ) ∪ (b ∩ (bc )))
3633, 35letr 137 . 2 a ≤ ((bc ) ∪ (b ∩ (bc )))
37 ax-a2 31 . . 3 ((bc ) ∪ (b ∩ (bc ))) = ((b ∩ (bc )) ∪ (bc ))
38 leao1 162 . . . 4 (b ∩ (bc )) ≤ (bc )
3938df-le2 131 . . 3 ((b ∩ (bc )) ∪ (bc )) = (bc )
4037, 39ax-r2 36 . 2 ((bc ) ∪ (b ∩ (bc ))) = (bc )
4136, 40lbtr 139 1 a ≤ (bc )
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12 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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  elimcons  868
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