Theorem ifnot 4535

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 143	. . . 4 ⊢ (𝜑 → ¬ ¬ 𝜑)
2	1	iffalsed 4493	. . 3 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵)
3		iftrue 4488	. . 3 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵)
4	2, 3	eqtr4d 2799	. 2 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
5		iftrue 4488	. . 3 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴)
6		iffalse 4491	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴)
7	5, 6	eqtr4d 2799	. 2 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
8	4, 7	pm2.61i 184	1 ⊢ if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ifcif 4482

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-if 4483

This theorem is used by:  suppsnop  8179  2resupmax  13299  sgnneg  15233  sadadd2lem2  16600  maducoeval2  22935  tmsxpsval2  24838  itg2uba  26044  lgsneg  27630  lgsdilem  27633  bj-xpimasn  37838  itgaddnclem2  38565  ftc1anclem5  38583