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Theorem ltlecasei 10736
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 10735 . . 3 ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵𝐴𝐴 < 𝐵))
63, 4, 5syl2anc 584 . 2 (𝜑 → (𝐵𝐴𝐴 < 𝐵))
71, 2, 6mpjaodan 952 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841  wcel 2105   class class class wbr 5057  cr 10524   < clt 10663  cle 10664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-xr 10667  df-le 10669
This theorem is referenced by:  iccsplit  12859  expnbnd  13581  hashf1  13803  absmax  14677  sinltx  15530  iccntr  23356  pmltpclem2  23977  cniccbdd  23989  iccvolcl  24095  ioovolcl  24098  dyaddisjlem  24123  mbfposr  24180  itg1ge0a  24239  itg2monolem1  24278  itgioo  24343  c1lip1  24521  plyeq0lem  24727  aalioulem5  24852  pserulm  24937  tanord  25049  birthdaylem3  25458  fsumharmonic  25516  chpo1ubb  25984  mblfinlem2  34811  ioodvbdlimc1  42094  ioodvbdlimc2  42096  ibliooicc  42132  fourierdlem107  42375
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