Theorem ifnot 3701

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		notnot1 114	. . . 4 ⊢ (φ → ¬ ¬ φ)
2		iffalse 3670	. . . 4 ⊢ (¬ ¬ φ → if(¬ φ, A, B) = B)
3	1, 2	syl 15	. . 3 ⊢ (φ → if(¬ φ, A, B) = B)
4		iftrue 3669	. . 3 ⊢ (φ → if(φ, B, A) = B)
5	3, 4	eqtr4d 2388	. 2 ⊢ (φ → if(¬ φ, A, B) = if(φ, B, A))
6		iftrue 3669	. . 3 ⊢ (¬ φ → if(¬ φ, A, B) = A)
7		iffalse 3670	. . 3 ⊢ (¬ φ → if(φ, B, A) = A)
8	6, 7	eqtr4d 2388	. 2 ⊢ (¬ φ → if(¬ φ, A, B) = if(φ, B, A))
9	5, 8	pm2.61i 156	1 ⊢ if(¬ φ, A, B) = if(φ, B, A)

Syntax hints:  ¬ wn 3   = wceq 1642   ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3664

This theorem is referenced by: (None)