Theorem ifnot 4514

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 4472	. . 3 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵)
3		iftrue 4467	. . 3 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵)
4	2, 3	eqtr4d 2778	. 2 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
5		iftrue 4467	. . 3 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴)
6		iffalse 4470	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴)
7	5, 6	eqtr4d 2778	. 2 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
8	4, 7	pm2.61i 183	1 ⊢ if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)

Syntax hints:  ¬ wn 3   = wceq 1547  ifcif 4461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-if 4462

This theorem is referenced by:  suppsnop  8125  2resupmax  13138  sadadd2lem2  16417  maducoeval2  22630  tmsxpsval2  24529  itg2uba  25735  lgsneg  27309  lgsdilem  27312  sgnneg  32932  bj-xpimasn  37315  itgaddnclem2  38053  ftc1anclem5  38071