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Theorem dfif3 3337
 Description: Alternate definition of the conditional operator df-if 3326. Note that φ is independent of x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)
Hypothesis
Ref Expression
dfif3.1 𝐶 = {xφ}
Assertion
Ref Expression
dfif3 if(φ, A, B) = ((A𝐶) ∪ (B ∩ (V ∖ 𝐶)))
Distinct variable group:   φ,x
Allowed substitution hints:   A(x)   B(x)   𝐶(x)

Proof of Theorem dfif3
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfif6 3327 . 2 if(φ, A, B) = ({y Aφ} ∪ {y B ∣ ¬ φ})
2 dfif3.1 . . . . . 6 𝐶 = {xφ}
3 biidd 161 . . . . . . 7 (x = y → (φφ))
43cbvabv 2158 . . . . . 6 {xφ} = {yφ}
52, 4eqtri 2057 . . . . 5 𝐶 = {yφ}
65ineq2i 3129 . . . 4 (A𝐶) = (A ∩ {yφ})
7 dfrab3 3207 . . . 4 {y Aφ} = (A ∩ {yφ})
86, 7eqtr4i 2060 . . 3 (A𝐶) = {y Aφ}
9 dfrab3 3207 . . . 4 {y B ∣ ¬ φ} = (B ∩ {y ∣ ¬ φ})
10 notab 3201 . . . . . 6 {y ∣ ¬ φ} = (V ∖ {yφ})
115difeq2i 3053 . . . . . 6 (V ∖ 𝐶) = (V ∖ {yφ})
1210, 11eqtr4i 2060 . . . . 5 {y ∣ ¬ φ} = (V ∖ 𝐶)
1312ineq2i 3129 . . . 4 (B ∩ {y ∣ ¬ φ}) = (B ∩ (V ∖ 𝐶))
149, 13eqtr2i 2058 . . 3 (B ∩ (V ∖ 𝐶)) = {y B ∣ ¬ φ}
158, 14uneq12i 3089 . 2 ((A𝐶) ∪ (B ∩ (V ∖ 𝐶))) = ({y Aφ} ∪ {y B ∣ ¬ φ})
161, 15eqtr4i 2060 1 if(φ, A, B) = ((A𝐶) ∪ (B ∩ (V ∖ 𝐶)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   = wceq 1242  {cab 2023  {crab 2304  Vcvv 2551   ∖ cdif 2908   ∪ cun 2909   ∩ cin 2910  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-in1 544  ax-in2 545  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-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-if 3326 This theorem is referenced by: (None)
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