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Theorem somin2 6136
Description: Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
somin2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐵)

Proof of Theorem somin2
StepHypRef Expression
1 somincom 6135 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴))
2 somin1 6134 . . 3 ((𝑅 Or 𝑋 ∧ (𝐵𝑋𝐴𝑋)) → if(𝐵𝑅𝐴, 𝐵, 𝐴)(𝑅 ∪ I )𝐵)
32ancom2s 648 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐵𝑅𝐴, 𝐵, 𝐴)(𝑅 ∪ I )𝐵)
41, 3eqbrtrd 5170 1 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  cun 3946  ifcif 4528   class class class wbr 5148   I cid 5573   Or wor 5587
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-3or 1088  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-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683
This theorem is referenced by:  soltmin  6137
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