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Theorem ltlecasei 10024
 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 10023 . . 3 ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵𝐴𝐴 < 𝐵))
63, 4, 5syl2anc 691 . 2 (𝜑 → (𝐵𝐴𝐴 < 𝐵))
71, 2, 6mpjaodan 823 1 (𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∈ wcel 1977   class class class wbr 4583  ℝcr 9814   < clt 9953   ≤ cle 9954 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-xr 9957  df-le 9959 This theorem is referenced by:  iccsplit  12176  expnbnd  12855  hashf1  13098  absmax  13917  sinltx  14758  iccntr  22432  pmltpclem2  23025  cniccbdd  23037  iccvolcl  23142  ioovolcl  23144  dyaddisjlem  23169  mbfposr  23225  itg1ge0a  23284  itg2monolem1  23323  itgioo  23388  c1lip1  23564  plyeq0lem  23770  aalioulem5  23895  pserulm  23980  tanord  24088  birthdaylem3  24480  fsumharmonic  24538  chpo1ubb  24970  mblfinlem2  32617  ioodvbdlimc1  38823  ioodvbdlimc2  38825  ibliooicc  38863  fourierdlem107  39106
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