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Theorem ifnot 4166
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 4130 . . 3 (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵)
3 iftrue 4125 . . 3 (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵)
42, 3eqtr4d 2688 . 2 (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
5 iftrue 4125 . . 3 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴)
6 iffalse 4128 . . 3 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴)
75, 6eqtr4d 2688 . 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 1523  ifcif 4119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-if 4120
This theorem is referenced by:  suppsnop  7354  2resupmax  12057  sadadd2lem2  15219  maducoeval2  20494  tmsxpsval2  22391  itg2uba  23555  lgsneg  25091  lgsdilem  25094  sgnneg  30730  bj-xpimasn  33067  itgaddnclem2  33599  ftc1anclem5  33619
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