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Theorem ltlecasei 11369
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 11368 . . 3 ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵𝐴𝐴 < 𝐵))
63, 4, 5syl2anc 584 . 2 (𝜑 → (𝐵𝐴𝐴 < 𝐵))
71, 2, 6mpjaodan 961 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848  wcel 2108   class class class wbr 5143  cr 11154   < clt 11295  cle 11296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-xr 11299  df-le 11301
This theorem is referenced by:  iccsplit  13525  expnbnd  14271  hashf1  14496  absmax  15368  sinltx  16225  iccntr  24843  pmltpclem2  25484  cniccbdd  25496  iccvolcl  25602  ioovolcl  25605  dyaddisjlem  25630  mbfposr  25687  itg1ge0a  25746  itg2monolem1  25785  itgioo  25851  c1lip1  26036  plyeq0lem  26249  aalioulem5  26378  pserulm  26465  tanord  26580  birthdaylem3  26996  fsumharmonic  27055  chpo1ubb  27525  mblfinlem2  37665  metakunt9  42214  ioodvbdlimc1  45948  ioodvbdlimc2  45950  ibliooicc  45986  fourierdlem107  46228
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