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Theorem ltlecasei 10435
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 10434 . . 3 ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵𝐴𝐴 < 𝐵))
63, 4, 5syl2anc 580 . 2 (𝜑 → (𝐵𝐴𝐴 < 𝐵))
71, 2, 6mpjaodan 982 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wo 874  wcel 2157   class class class wbr 4843  cr 10223   < clt 10363  cle 10364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-cnv 5320  df-xr 10367  df-le 10369
This theorem is referenced by:  iccsplit  12559  expnbnd  13247  hashf1  13490  absmax  14410  sinltx  15255  iccntr  22952  pmltpclem2  23557  cniccbdd  23569  iccvolcl  23675  ioovolcl  23678  dyaddisjlem  23703  mbfposr  23760  itg1ge0a  23819  itg2monolem1  23858  itgioo  23923  c1lip1  24101  plyeq0lem  24307  aalioulem5  24432  pserulm  24517  tanord  24626  birthdaylem3  25032  fsumharmonic  25090  chpo1ubb  25522  mblfinlem2  33936  ioodvbdlimc1  40892  ioodvbdlimc2  40894  ibliooicc  40930  fourierdlem107  41173
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