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Theorem ltlecasei 10725
Description: Ordering elimination by cases. (Contributed by NM, 1-Jul-2007.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
ltlecasei.1 ((𝜑𝐴 < 𝐵) → 𝜓)
ltlecasei.2 ((𝜑𝐵𝐴) → 𝜓)
ltlecasei.3 (𝜑𝐴 ∈ ℝ)
ltlecasei.4 (𝜑𝐵 ∈ ℝ)
Assertion
Ref Expression
ltlecasei (𝜑𝜓)

Proof of Theorem ltlecasei
StepHypRef Expression
1 ltlecasei.2 . 2 ((𝜑𝐵𝐴) → 𝜓)
2 ltlecasei.1 . 2 ((𝜑𝐴 < 𝐵) → 𝜓)
3 ltlecasei.4 . . 3 (𝜑𝐵 ∈ ℝ)
4 ltlecasei.3 . . 3 (𝜑𝐴 ∈ ℝ)
5 lelttric 10724 . . 3 ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵𝐴𝐴 < 𝐵))
63, 4, 5syl2anc 587 . 2 (𝜑 → (𝐵𝐴𝐴 < 𝐵))
71, 2, 6mpjaodan 956 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  wcel 2115   class class class wbr 5039  cr 10513   < clt 10652  cle 10653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-xp 5534  df-cnv 5536  df-xr 10656  df-le 10658
This theorem is referenced by:  iccsplit  12853  expnbnd  13577  hashf1  13799  absmax  14668  sinltx  15521  iccntr  23405  pmltpclem2  24032  cniccbdd  24044  iccvolcl  24150  ioovolcl  24153  dyaddisjlem  24178  mbfposr  24235  itg1ge0a  24294  itg2monolem1  24333  itgioo  24398  c1lip1  24579  plyeq0lem  24786  aalioulem5  24911  pserulm  24996  tanord  25109  birthdaylem3  25518  fsumharmonic  25576  chpo1ubb  26044  mblfinlem2  34977  ioodvbdlimc1  42394  ioodvbdlimc2  42396  ibliooicc  42432  fourierdlem107  42674
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