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Theorem somin2 5450
 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 5449 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴))
2 somin1 5448 . . 3 ((𝑅 Or 𝑋 ∧ (𝐵𝑋𝐴𝑋)) → if(𝐵𝑅𝐴, 𝐵, 𝐴)(𝑅 ∪ I )𝐵)
32ancom2s 840 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐵𝑅𝐴, 𝐵, 𝐴)(𝑅 ∪ I )𝐵)
41, 3eqbrtrd 4605 1 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ∪ cun 3538  ifcif 4036   class class class wbr 4583   I cid 4948   Or wor 4958 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045 This theorem is referenced by:  soltmin  5451
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