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

Theorem ifbid 3680
 Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
Hypothesis
Ref Expression
ifbid.1 (φ → (ψχ))
Assertion
Ref Expression
ifbid (φ → if(ψ, A, B) = if(χ, A, B))

Proof of Theorem ifbid
StepHypRef Expression
1 ifbid.1 . 2 (φ → (ψχ))
2 ifbi 3679 . 2 ((ψχ) → if(ψ, A, B) = if(χ, A, B))
31, 2syl 15 1 (φ → if(ψ, A, B) = if(χ, A, B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ifcif 3662 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  ifbieq2d  3682  ifbieq12d  3684  ifan  3701  ifor  3702  enprmaplem5  6080
 Copyright terms: Public domain W3C validator