Theorem ifnot 4558

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 4516	. . 3 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵)
3		iftrue 4511	. . 3 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵)
4	2, 3	eqtr4d 2774	. 2 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
5		iftrue 4511	. . 3 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴)
6		iffalse 4514	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴)
7	5, 6	eqtr4d 2774	. 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 1540  ifcif 4505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-if 4506

This theorem is referenced by:  suppsnop  8182  2resupmax  13209  sadadd2lem2  16474  maducoeval2  22583  tmsxpsval2  24483  itg2uba  25701  lgsneg  27289  lgsdilem  27292  sgnneg  32817  bj-xpimasn  36978  itgaddnclem2  37708  ftc1anclem5  37726