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Theorem ltlecasei 11321
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 11320 . . 3 ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵𝐴𝐴 < 𝐵))
63, 4, 5syl2anc 584 . 2 (𝜑 → (𝐵𝐴𝐴 < 𝐵))
71, 2, 6mpjaodan 957 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845  wcel 2106   class class class wbr 5148  cr 11108   < clt 11247  cle 11248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-xr 11251  df-le 11253
This theorem is referenced by:  iccsplit  13461  expnbnd  14194  hashf1  14417  absmax  15275  sinltx  16131  iccntr  24336  pmltpclem2  24965  cniccbdd  24977  iccvolcl  25083  ioovolcl  25086  dyaddisjlem  25111  mbfposr  25168  itg1ge0a  25228  itg2monolem1  25267  itgioo  25332  c1lip1  25513  plyeq0lem  25723  aalioulem5  25848  pserulm  25933  tanord  26046  birthdaylem3  26455  fsumharmonic  26513  chpo1ubb  26981  mblfinlem2  36521  metakunt9  40988  ioodvbdlimc1  44639  ioodvbdlimc2  44641  ibliooicc  44677  fourierdlem107  44919
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