Theorem ifnot 4580

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 4539	. . 3 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵)
3		iftrue 4534	. . 3 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵)
4	2, 3	eqtr4d 2775	. 2 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
5		iftrue 4534	. . 3 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴)
6		iffalse 4537	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴)
7	5, 6	eqtr4d 2775	. 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 1541  ifcif 4528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-if 4529

This theorem is referenced by:  suppsnop  8162  2resupmax  13166  sadadd2lem2  16390  maducoeval2  22141  tmsxpsval2  24047  itg2uba  25260  lgsneg  26821  lgsdilem  26824  sgnneg  33534  bj-xpimasn  35831  itgaddnclem2  36542  ftc1anclem5  36560