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Theorem dfif6 3665
 Description: An alternate definition of the conditional operator df-if 3663 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
dfif6 if(φ, A, B) = ({x A φ} ∪ {x B ¬ φ})
Distinct variable groups:   φ,x   x,A   x,B

Proof of Theorem dfif6
StepHypRef Expression
1 unab 3521 . 2 ({x (x A φ)} ∪ {x (x B ¬ φ)}) = {x ((x A φ) (x B ¬ φ))}
2 df-rab 2623 . . 3 {x A φ} = {x (x A φ)}
3 df-rab 2623 . . 3 {x B ¬ φ} = {x (x B ¬ φ)}
42, 3uneq12i 3416 . 2 ({x A φ} ∪ {x B ¬ φ}) = ({x (x A φ)} ∪ {x (x B ¬ φ)})
5 df-if 3663 . 2 if(φ, A, B) = {x ((x A φ) (x B ¬ φ))}
61, 4, 53eqtr4ri 2384 1 if(φ, A, B) = ({x A φ} ∪ {x B ¬ φ})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {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:  ifeq1  3666  ifeq2  3667  dfif3  3672
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