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Theorem ltlecasei 10346
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 10345 . . 3 ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵𝐴𝐴 < 𝐵))
63, 4, 5syl2anc 565 . 2 (𝜑 → (𝐵𝐴𝐴 < 𝐵))
71, 2, 6mpjaodan 949 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 826  wcel 2145   class class class wbr 4786  cr 10136   < clt 10275  cle 10276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-xr 10279  df-le 10281
This theorem is referenced by:  iccsplit  12511  expnbnd  13199  hashf1  13442  absmax  14276  sinltx  15124  iccntr  22843  pmltpclem2  23436  cniccbdd  23448  iccvolcl  23554  ioovolcl  23557  dyaddisjlem  23582  mbfposr  23638  itg1ge0a  23697  itg2monolem1  23736  itgioo  23801  c1lip1  23979  plyeq0lem  24185  aalioulem5  24310  pserulm  24395  tanord  24504  birthdaylem3  24900  fsumharmonic  24958  chpo1ubb  25390  mblfinlem2  33776  ioodvbdlimc1  40662  ioodvbdlimc2  40664  ibliooicc  40700  fourierdlem107  40943
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