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Theorem ifnot 4082
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 134 . . . 4 (𝜑 → ¬ ¬ 𝜑)
21iffalsed 4046 . . 3 (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵)
3 iftrue 4041 . . 3 (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵)
42, 3eqtr4d 2646 . 2 (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
5 iftrue 4041 . . 3 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴)
6 iffalse 4044 . . 3 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴)
75, 6eqtr4d 2646 . 2 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
84, 7pm2.61i 174 1 if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1474  ifcif 4035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-if 4036
This theorem is referenced by:  suppsnop  7173  2resupmax  11852  sadadd2lem2  14956  maducoeval2  20207  tmsxpsval2  22095  itg2uba  23233  lgsneg  24763  lgsdilem  24766  sgnneg  29735  bj-xpimasn  31931  itgaddnclem2  32435  ftc1anclem5  32455
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