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Theorem somincom 5746
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 5255 . . . . 5 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ¬ (𝐴𝑅𝐵𝐵𝑅𝐴))
2 nan 860 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ¬ (𝐴𝑅𝐵𝐵𝑅𝐴)) ↔ (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → ¬ 𝐵𝑅𝐴))
31, 2mpbi 222 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → ¬ 𝐵𝑅𝐴)
43iffalsed 4286 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐴)
54eqcomd 2803 . 2 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → 𝐴 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
6 sotric 5257 . . . . 5 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝑅𝐴)))
76con2bid 346 . . . 4 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴 = 𝐵𝐵𝑅𝐴) ↔ ¬ 𝐴𝑅𝐵))
8 ifeq2 4280 . . . . . 6 (𝐴 = 𝐵 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = if(𝐵𝑅𝐴, 𝐵, 𝐵))
9 ifid 4314 . . . . . 6 if(𝐵𝑅𝐴, 𝐵, 𝐵) = 𝐵
108, 9syl6req 2848 . . . . 5 (𝐴 = 𝐵𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
11 iftrue 4281 . . . . . 6 (𝐵𝑅𝐴 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐵)
1211eqcomd 2803 . . . . 5 (𝐵𝑅𝐴𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
1310, 12jaoi 884 . . . 4 ((𝐴 = 𝐵𝐵𝑅𝐴) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
147, 13syl6bir 246 . . 3 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (¬ 𝐴𝑅𝐵𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴)))
1514imp 396 . 2 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝑅𝐵) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
165, 15ifeqda 4310 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 385  wo 874   = wceq 1653  wcel 2157  ifcif 4275   class class class wbr 4841   Or wor 5230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-po 5231  df-so 5232
This theorem is referenced by:  somin2  5747
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