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Theorem ifnot 3700
 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
StepHypRef Expression
1 notnot1 114 . . . 4 (φ → ¬ ¬ φ)
2 iffalse 3669 . . . 4 (¬ ¬ φ → if(¬ φ, A, B) = B)
31, 2syl 15 . . 3 (φ → if(¬ φ, A, B) = B)
4 iftrue 3668 . . 3 (φ → if(φ, B, A) = B)
53, 4eqtr4d 2388 . 2 (φ → if(¬ φ, A, B) = if(φ, B, A))
6 iftrue 3668 . . 3 φ → if(¬ φ, A, B) = A)
7 iffalse 3669 . . 3 φ → if(φ, B, A) = A)
86, 7eqtr4d 2388 . 2 φ → if(¬ φ, A, B) = if(φ, B, A))
95, 8pm2.61i 156 1 if(¬ φ, A, B) = if(φ, B, A)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1642   ifcif 3662 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 3663 This theorem is referenced by: (None)
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