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Theorem ltlecasei 11326
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 11325 . . 3 ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵𝐴𝐴 < 𝐵))
63, 4, 5syl2anc 582 . 2 (𝜑 → (𝐵𝐴𝐴 < 𝐵))
71, 2, 6mpjaodan 955 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 843  wcel 2104   class class class wbr 5147  cr 11111   < clt 11252  cle 11253
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-xr 11256  df-le 11258
This theorem is referenced by:  iccsplit  13466  expnbnd  14199  hashf1  14422  absmax  15280  sinltx  16136  iccntr  24557  pmltpclem2  25198  cniccbdd  25210  iccvolcl  25316  ioovolcl  25319  dyaddisjlem  25344  mbfposr  25401  itg1ge0a  25461  itg2monolem1  25500  itgioo  25565  c1lip1  25749  plyeq0lem  25959  aalioulem5  26085  pserulm  26170  tanord  26283  birthdaylem3  26694  fsumharmonic  26752  chpo1ubb  27220  mblfinlem2  36829  metakunt9  41299  ioodvbdlimc1  44947  ioodvbdlimc2  44949  ibliooicc  44985  fourierdlem107  45227
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