MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ifnot Structured version   Visualization version   GIF version

Theorem ifnot 4490
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 144 . . . 4 (𝜑 → ¬ ¬ 𝜑)
21iffalsed 4451 . . 3 (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵)
3 iftrue 4446 . . 3 (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵)
42, 3eqtr4d 2859 . 2 (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
5 iftrue 4446 . . 3 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴)
6 iffalse 4449 . . 3 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴)
75, 6eqtr4d 2859 . 2 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
84, 7pm2.61i 185 1 if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  ifcif 4440
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-if 4441
This theorem is referenced by:  suppsnop  7819  2resupmax  12559  sadadd2lem2  15776  maducoeval2  21224  tmsxpsval2  23124  itg2uba  24325  lgsneg  25883  lgsdilem  25886  sgnneg  31805  bj-xpimasn  34283  itgaddnclem2  34996  ftc1anclem5  35014
  Copyright terms: Public domain W3C validator