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Theorem somincom 5432
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 4969 . . . . 5 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ¬ (𝐴𝑅𝐵𝐵𝑅𝐴))
2 nan 601 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ¬ (𝐴𝑅𝐵𝐵𝑅𝐴)) ↔ (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → ¬ 𝐵𝑅𝐴))
31, 2mpbi 218 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → ¬ 𝐵𝑅𝐴)
43iffalsed 4042 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐴)
54eqcomd 2611 . 2 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → 𝐴 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
6 sotric 4971 . . . . 5 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝑅𝐴)))
76con2bid 342 . . . 4 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴 = 𝐵𝐵𝑅𝐴) ↔ ¬ 𝐴𝑅𝐵))
8 ifeq2 4036 . . . . . 6 (𝐴 = 𝐵 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = if(𝐵𝑅𝐴, 𝐵, 𝐵))
9 ifid 4070 . . . . . 6 if(𝐵𝑅𝐴, 𝐵, 𝐵) = 𝐵
108, 9syl6req 2656 . . . . 5 (𝐴 = 𝐵𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
11 iftrue 4037 . . . . . 6 (𝐵𝑅𝐴 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐵)
1211eqcomd 2611 . . . . 5 (𝐵𝑅𝐴𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
1310, 12jaoi 392 . . . 4 ((𝐴 = 𝐵𝐵𝑅𝐴) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
147, 13syl6bir 242 . . 3 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (¬ 𝐴𝑅𝐵𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴)))
1514imp 443 . 2 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝑅𝐵) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
165, 15ifeqda 4066 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  wo 381  wa 382   = wceq 1474  wcel 1975  ifcif 4031   class class class wbr 4573   Or wor 4944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-po 4945  df-so 4946
This theorem is referenced by:  somin2  5433
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