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

Theorem ifsb 3671
 Description: Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)
Hypotheses
Ref Expression
ifsb.1 ( if(φ, A, B) = AC = D)
ifsb.2 ( if(φ, A, B) = BC = E)
Assertion
Ref Expression
ifsb C = if(φ, D, E)

Proof of Theorem ifsb
StepHypRef Expression
1 iftrue 3668 . . . 4 (φ → if(φ, A, B) = A)
2 ifsb.1 . . . 4 ( if(φ, A, B) = AC = D)
31, 2syl 15 . . 3 (φC = D)
4 iftrue 3668 . . 3 (φ → if(φ, D, E) = D)
53, 4eqtr4d 2388 . 2 (φC = if(φ, D, E))
6 iffalse 3669 . . . 4 φ → if(φ, A, B) = B)
7 ifsb.2 . . . 4 ( if(φ, A, B) = BC = E)
86, 7syl 15 . . 3 φC = E)
9 iffalse 3669 . . 3 φ → if(φ, D, E) = E)
108, 9eqtr4d 2388 . 2 φC = if(φ, D, E))
115, 10pm2.61i 156 1 C = if(φ, D, E)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = 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:  fvif  5340
 Copyright terms: Public domain W3C validator