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Theorem dfif6 4310
 Description: An alternate definition of the conditional operator df-if 4308 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 4120 . 2 ({𝑥 ∣ (𝑥𝐴𝜑)} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ¬ 𝜑)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
2 df-rab 3099 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
3 df-rab 3099 . . 3 {𝑥𝐵 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥𝐵 ∧ ¬ 𝜑)}
42, 3uneq12i 3988 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ¬ 𝜑)})
5 df-if 4308 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
61, 4, 53eqtr4ri 2813 1 if(𝜑, 𝐴, 𝐵) = ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 386   ∨ wo 836   = wceq 1601   ∈ wcel 2107  {cab 2763  {crab 3094   ∪ cun 3790  ifcif 4307 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-un 3797  df-if 4308 This theorem is referenced by:  ifeq1  4311  ifeq2  4312  dfif3  4321
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