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Theorem dfif6 3327
Description: An alternate definition of the conditional operator df-if 3326 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
dfif6 if(φ, A, B) = ({x Aφ} ∪ {x B ∣ ¬ φ})
Distinct variable groups:   φ,x   x,A   x,B

Proof of Theorem dfif6
StepHypRef Expression
1 unab 3198 . 2 ({x ∣ (x A φ)} ∪ {x ∣ (x B ¬ φ)}) = {x ∣ ((x A φ) (x B ¬ φ))}
2 df-rab 2309 . . 3 {x Aφ} = {x ∣ (x A φ)}
3 df-rab 2309 . . 3 {x B ∣ ¬ φ} = {x ∣ (x B ¬ φ)}
42, 3uneq12i 3089 . 2 ({x Aφ} ∪ {x B ∣ ¬ φ}) = ({x ∣ (x A φ)} ∪ {x ∣ (x B ¬ φ)})
5 df-if 3326 . 2 if(φ, A, B) = {x ∣ ((x A φ) (x B ¬ φ))}
61, 4, 53eqtr4ri 2068 1 if(φ, A, B) = ({x Aφ} ∪ {x B ∣ ¬ φ})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   wo 628   = wceq 1242   wcel 1390  {cab 2023  {crab 2304  cun 2909  ifcif 3325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-un 2916  df-if 3326
This theorem is referenced by:  ifeq1  3328  ifeq2  3329  dfif3  3337
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