New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  ifel GIF version

Theorem ifel 3697
 Description: Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
Assertion
Ref Expression
ifel ( if(φ, A, B) C ↔ ((φ A C) φ B C)))

Proof of Theorem ifel
StepHypRef Expression
1 eleq1 2413 . 2 ( if(φ, A, B) = A → ( if(φ, A, B) CA C))
2 eleq1 2413 . 2 ( if(φ, A, B) = B → ( if(φ, A, B) CB C))
31, 2elimif 3691 1 ( if(φ, A, B) C ↔ ((φ A C) φ B C)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∈ wcel 1710   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)
 Copyright terms: Public domain W3C validator