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Theorem elimcons 868
 Description: Consequent elimination law.
Hypotheses
Ref Expression
elimcons.1 (a1 c) = (b1 c)
elimcons.2 (ac) ≤ (bc )
Assertion
Ref Expression
elimcons ab

Proof of Theorem elimcons
StepHypRef Expression
1 df-t 41 . . . . . . . 8 1 = (aa )
2 elimcons.1 . . . . . . . . . 10 (a1 c) = (b1 c)
3 elimcons.2 . . . . . . . . . 10 (ac) ≤ (bc )
42, 3elimconslem 867 . . . . . . . . 9 a ≤ (bc )
54leror 152 . . . . . . . 8 (aa ) ≤ ((bc ) ∪ a )
61, 5bltr 138 . . . . . . 7 1 ≤ ((bc ) ∪ a )
76lelan 167 . . . . . 6 (b ∩ 1) ≤ (b ∩ ((bc ) ∪ a ))
8 an1 106 . . . . . 6 (b ∩ 1) = b
9 comor1 461 . . . . . . . 8 (bc ) C b
109comcom2 183 . . . . . . 7 (bc ) C b
114lecom 180 . . . . . . . . 9 a C (bc )
1211comcom3 454 . . . . . . . 8 a C (bc )
1312comcom 453 . . . . . . 7 (bc ) C a
1410, 13fh2 470 . . . . . 6 (b ∩ ((bc ) ∪ a )) = ((b ∩ (bc )) ∪ (ba ))
157, 8, 14le3tr2 141 . . . . 5 b ≤ ((b ∩ (bc )) ∪ (ba ))
162negant 852 . . . . . . . . . . 11 (a1 c) = (b1 c)
17 df-i1 44 . . . . . . . . . . 11 (a1 c) = (a ∪ (ac))
18 df-i1 44 . . . . . . . . . . 11 (b1 c) = (b ∪ (bc))
1916, 17, 183tr2 64 . . . . . . . . . 10 (a ∪ (ac)) = (b ∪ (bc))
20 anor2 89 . . . . . . . . . . 11 (ac) = (ac )
2120lor 70 . . . . . . . . . 10 (a ∪ (ac)) = (a ∪ (ac ) )
22 anor2 89 . . . . . . . . . . 11 (bc) = (bc )
2322lor 70 . . . . . . . . . 10 (b ∪ (bc)) = (b ∪ (bc ) )
2419, 21, 233tr2 64 . . . . . . . . 9 (a ∪ (ac ) ) = (b ∪ (bc ) )
2524ax-r1 35 . . . . . . . 8 (b ∪ (bc ) ) = (a ∪ (ac ) )
2625ax-r4 37 . . . . . . 7 (b ∪ (bc ) ) = (a ∪ (ac ) )
27 df-a 40 . . . . . . 7 (b ∩ (bc )) = (b ∪ (bc ) )
28 df-a 40 . . . . . . 7 (a ∩ (ac )) = (a ∪ (ac ) )
2926, 27, 283tr1 63 . . . . . 6 (b ∩ (bc )) = (a ∩ (ac ))
3029ax-r5 38 . . . . 5 ((b ∩ (bc )) ∪ (ba )) = ((a ∩ (ac )) ∪ (ba ))
3115, 30lbtr 139 . . . 4 b ≤ ((a ∩ (ac )) ∪ (ba ))
32 lear 161 . . . . 5 (ba ) ≤ a
3332lelor 166 . . . 4 ((a ∩ (ac )) ∪ (ba )) ≤ ((a ∩ (ac )) ∪ a )
3431, 33letr 137 . . 3 b ≤ ((a ∩ (ac )) ∪ a )
35 lea 160 . . . 4 (a ∩ (ac )) ≤ a
3635df-le2 131 . . 3 ((a ∩ (ac )) ∪ a ) = a
3734, 36lbtr 139 . 2 ba
3837lecon1 155 1 ab
 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:  elimcons2  869
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