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Theorem ifnot 4277
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 136 . . . 4 (𝜑 → ¬ ¬ 𝜑)
21iffalsed 4241 . . 3 (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵)
3 iftrue 4236 . . 3 (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵)
42, 3eqtr4d 2797 . 2 (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
5 iftrue 4236 . . 3 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴)
6 iffalse 4239 . . 3 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴)
75, 6eqtr4d 2797 . 2 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
84, 7pm2.61i 176 1 if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  ifcif 4230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-if 4231
This theorem is referenced by:  suppsnop  7478  2resupmax  12232  sadadd2lem2  15394  maducoeval2  20668  tmsxpsval2  22565  itg2uba  23729  lgsneg  25266  lgsdilem  25269  sgnneg  30932  bj-xpimasn  33266  itgaddnclem2  33800  ftc1anclem5  33820
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