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Theorem somincom 6135
Description: Commutativity of minimum in a total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
somincom ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴))

Proof of Theorem somincom
StepHypRef Expression
1 so2nr 5614 . . . . 5 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ¬ (𝐴𝑅𝐵𝐵𝑅𝐴))
2 nan 828 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ¬ (𝐴𝑅𝐵𝐵𝑅𝐴)) ↔ (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → ¬ 𝐵𝑅𝐴))
31, 2mpbi 229 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → ¬ 𝐵𝑅𝐴)
43iffalsed 4539 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐴)
54eqcomd 2738 . 2 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → 𝐴 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
6 sotric 5616 . . . . 5 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝑅𝐴)))
76con2bid 354 . . . 4 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴 = 𝐵𝐵𝑅𝐴) ↔ ¬ 𝐴𝑅𝐵))
8 ifeq2 4533 . . . . . 6 (𝐴 = 𝐵 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = if(𝐵𝑅𝐴, 𝐵, 𝐵))
9 ifid 4568 . . . . . 6 if(𝐵𝑅𝐴, 𝐵, 𝐵) = 𝐵
108, 9eqtr2di 2789 . . . . 5 (𝐴 = 𝐵𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
11 iftrue 4534 . . . . . 6 (𝐵𝑅𝐴 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐵)
1211eqcomd 2738 . . . . 5 (𝐵𝑅𝐴𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
1310, 12jaoi 855 . . . 4 ((𝐴 = 𝐵𝐵𝑅𝐴) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
147, 13syl6bir 253 . . 3 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (¬ 𝐴𝑅𝐵𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴)))
1514imp 407 . 2 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝑅𝐵) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
165, 15ifeqda 4564 1 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845   = wceq 1541  wcel 2106  ifcif 4528   class class class wbr 5148   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
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-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-po 5588  df-so 5589
This theorem is referenced by:  somin2  6136
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