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Theorem dfif2 3664
 Description: An alternate definition of the conditional operator df-if 3663 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)
Assertion
Ref Expression
dfif2 if(φ, A, B) = {x ((x Bφ) → (x A φ))}
Distinct variable groups:   φ,x   x,A   x,B

Proof of Theorem dfif2
StepHypRef Expression
1 df-if 3663 . 2 if(φ, A, B) = {x ((x A φ) (x B ¬ φ))}
2 df-or 359 . . . 4 (((x B ¬ φ) (x A φ)) ↔ (¬ (x B ¬ φ) → (x A φ)))
3 orcom 376 . . . 4 (((x A φ) (x B ¬ φ)) ↔ ((x B ¬ φ) (x A φ)))
4 iman 413 . . . . 5 ((x Bφ) ↔ ¬ (x B ¬ φ))
54imbi1i 315 . . . 4 (((x Bφ) → (x A φ)) ↔ (¬ (x B ¬ φ) → (x A φ)))
62, 3, 53bitr4i 268 . . 3 (((x A φ) (x B ¬ φ)) ↔ ((x Bφ) → (x A φ)))
76abbii 2465 . 2 {x ((x A φ) (x B ¬ φ))} = {x ((x Bφ) → (x A φ))}
81, 7eqtri 2373 1 if(φ, A, B) = {x ((x Bφ) → (x A φ))}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339   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-if 3663 This theorem is referenced by:  iftrue  3668  nfifd  3685
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