Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  elimif Structured version   Visualization version   GIF version

Theorem elimif 4457
 Description: Elimination of a conditional operator contained in a wff 𝜓. (Contributed by NM, 15-Feb-2005.) (Proof shortened by NM, 25-Apr-2019.)
Hypotheses
Ref Expression
elimif.1 (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝜓𝜒))
elimif.2 (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝜓𝜃))
Assertion
Ref Expression
elimif (𝜓 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))

Proof of Theorem elimif
StepHypRef Expression
1 iftrue 4426 . . 3 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2 elimif.1 . . 3 (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝜓𝜒))
31, 2syl 17 . 2 (𝜑 → (𝜓𝜒))
4 iffalse 4429 . . 3 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
5 elimif.2 . . 3 (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝜓𝜃))
64, 5syl 17 . 2 𝜑 → (𝜓𝜃))
73, 6cases 1038 1 (𝜓 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538  ifcif 4420 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-if 4421 This theorem is referenced by:  eqif  4461  elif  4463  ifel  4464  ftc1anclem5  35414  clsk1indlem2  41118
 Copyright terms: Public domain W3C validator