New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  ifeq2 GIF version

Theorem ifeq2 3667
 Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
ifeq2 (A = B → if(φ, C, A) = if(φ, C, B))

Proof of Theorem ifeq2
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 rabeq 2853 . . 3 (A = B → {x A ¬ φ} = {x B ¬ φ})
21uneq2d 3418 . 2 (A = B → ({x C φ} ∪ {x A ¬ φ}) = ({x C φ} ∪ {x B ¬ φ}))
3 dfif6 3665 . 2 if(φ, C, A) = ({x C φ} ∪ {x A ¬ φ})
4 dfif6 3665 . 2 if(φ, C, B) = ({x C φ} ∪ {x B ¬ φ})
52, 3, 43eqtr4g 2410 1 (A = B → if(φ, C, A) = if(φ, C, B))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642  {crab 2618   ∪ cun 3207   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-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-if 3663 This theorem is referenced by:  ifeq12  3675  ifeq2d  3677  ifbieq2i  3681  ifexg  3721
 Copyright terms: Public domain W3C validator