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Theorem ifnot 4581
Description: Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
Assertion
Ref Expression
ifnot if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)

Proof of Theorem ifnot
StepHypRef Expression
1 notnot 142 . . . 4 (𝜑 → ¬ ¬ 𝜑)
21iffalsed 4540 . . 3 (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵)
3 iftrue 4535 . . 3 (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵)
42, 3eqtr4d 2771 . 2 (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
5 iftrue 4535 . . 3 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴)
6 iffalse 4538 . . 3 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴)
75, 6eqtr4d 2771 . 2 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
84, 7pm2.61i 182 1 if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  ifcif 4529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-if 4530
This theorem is referenced by:  suppsnop  8182  2resupmax  13199  sadadd2lem2  16424  maducoeval2  22541  tmsxpsval2  24447  itg2uba  25672  lgsneg  27253  lgsdilem  27256  sgnneg  34160  bj-xpimasn  36434  itgaddnclem2  37152  ftc1anclem5  37170
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