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Theorem dfif6 4229
Description: An alternate definition of the conditional operator df-if 4227 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 4043 . 2 ({𝑥 ∣ (𝑥𝐴𝜑)} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ¬ 𝜑)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
2 df-rab 3070 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
3 df-rab 3070 . . 3 {𝑥𝐵 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥𝐵 ∧ ¬ 𝜑)}
42, 3uneq12i 3917 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ¬ 𝜑)})
5 df-if 4227 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
61, 4, 53eqtr4ri 2804 1 if(𝜑, 𝐴, 𝐵) = ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382  wo 828   = wceq 1631  wcel 2145  {cab 2757  {crab 3065  cun 3722  ifcif 4226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-un 3729  df-if 4227
This theorem is referenced by:  ifeq1  4230  ifeq2  4231  dfif3  4240
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