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Theorem somincom 5968
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 5468 . . . . 5 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ¬ (𝐴𝑅𝐵𝐵𝑅𝐴))
2 nan 829 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ¬ (𝐴𝑅𝐵𝐵𝑅𝐴)) ↔ (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → ¬ 𝐵𝑅𝐴))
31, 2mpbi 233 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → ¬ 𝐵𝑅𝐴)
43iffalsed 4425 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐴)
54eqcomd 2744 . 2 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝑅𝐵) → 𝐴 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
6 sotric 5470 . . . . 5 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝑅𝐴)))
76con2bid 358 . . . 4 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴 = 𝐵𝐵𝑅𝐴) ↔ ¬ 𝐴𝑅𝐵))
8 ifeq2 4419 . . . . . 6 (𝐴 = 𝐵 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = if(𝐵𝑅𝐴, 𝐵, 𝐵))
9 ifid 4454 . . . . . 6 if(𝐵𝑅𝐴, 𝐵, 𝐵) = 𝐵
108, 9eqtr2di 2790 . . . . 5 (𝐴 = 𝐵𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
11 iftrue 4420 . . . . . 6 (𝐵𝑅𝐴 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐵)
1211eqcomd 2744 . . . . 5 (𝐵𝑅𝐴𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
1310, 12jaoi 856 . . . 4 ((𝐴 = 𝐵𝐵𝑅𝐴) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
147, 13syl6bir 257 . . 3 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (¬ 𝐴𝑅𝐵𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴)))
1514imp 410 . 2 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝑅𝐵) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
165, 15ifeqda 4450 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 399  wo 846   = wceq 1542  wcel 2114  ifcif 4414   class class class wbr 5030   Or wor 5441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rab 3062  df-v 3400  df-un 3848  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-po 5442  df-so 5443
This theorem is referenced by:  somin2  5969
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