Theorem ifnot 4511

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

Step	Hyp	Ref	Expression
1		notnot 142	. . . 4 ⊢ (𝜑 → ¬ ¬ 𝜑)
2	1	iffalsed 4470	. . 3 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵)
3		iftrue 4465	. . 3 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵)
4	2, 3	eqtr4d 2781	. 2 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
5		iftrue 4465	. . 3 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴)
6		iffalse 4468	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴)
7	5, 6	eqtr4d 2781	. 2 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
8	4, 7	pm2.61i 182	1 ⊢ if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)

Syntax hints:  ¬ wn 3   = wceq 1539  ifcif 4459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-if 4460

This theorem is referenced by:  suppsnop  7994  2resupmax  12922  sadadd2lem2  16157  maducoeval2  21789  tmsxpsval2  23695  itg2uba  24908  lgsneg  26469  lgsdilem  26472  sgnneg  32507  bj-xpimasn  35145  itgaddnclem2  35836  ftc1anclem5  35854