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Theorem ltlecasei 10090
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 10089 . . 3 ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵𝐴𝐴 < 𝐵))
63, 4, 5syl2anc 692 . 2 (𝜑 → (𝐵𝐴𝐴 < 𝐵))
71, 2, 6mpjaodan 826 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384  wcel 1992   class class class wbr 4618  cr 9880   < clt 10019  cle 10020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087  df-xr 10023  df-le 10025
This theorem is referenced by:  iccsplit  12244  expnbnd  12930  hashf1  13176  absmax  13998  sinltx  14839  iccntr  22527  pmltpclem2  23120  cniccbdd  23132  iccvolcl  23237  ioovolcl  23239  dyaddisjlem  23264  mbfposr  23320  itg1ge0a  23379  itg2monolem1  23418  itgioo  23483  c1lip1  23659  plyeq0lem  23865  aalioulem5  23990  pserulm  24075  tanord  24183  birthdaylem3  24575  fsumharmonic  24633  chpo1ubb  25065  mblfinlem2  33065  ioodvbdlimc1  39441  ioodvbdlimc2  39443  ibliooicc  39481  fourierdlem107  39724
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