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Theorem dfif6 3381
Description: An alternate definition of the conditional operator df-if 3380 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
dfif6 if(𝜑, 𝐴, 𝐵) = ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfif6
StepHypRef Expression
1 unab 3255 . 2 ({𝑥 ∣ (𝑥𝐴𝜑)} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ¬ 𝜑)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
2 df-rab 2364 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
3 df-rab 2364 . . 3 {𝑥𝐵 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥𝐵 ∧ ¬ 𝜑)}
42, 3uneq12i 3141 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ¬ 𝜑)})
5 df-if 3380 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
61, 4, 53eqtr4ri 2116 1 if(𝜑, 𝐴, 𝐵) = ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wo 662   = wceq 1287  wcel 1436  {cab 2071  {crab 2359  cun 2986  ifcif 3379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rab 2364  df-v 2617  df-un 2992  df-if 3380
This theorem is referenced by:  ifeq1  3382  ifeq2  3383  dfif3  3392
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