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Theorem dfif6 3551
Description: An alternate definition of the conditional operator df-if 3550 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 3417 . 2 ({𝑥 ∣ (𝑥𝐴𝜑)} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ¬ 𝜑)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
2 df-rab 2477 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
3 df-rab 2477 . . 3 {𝑥𝐵 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥𝐵 ∧ ¬ 𝜑)}
42, 3uneq12i 3302 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ¬ 𝜑)})
5 df-if 3550 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
61, 4, 53eqtr4ri 2221 1 if(𝜑, 𝐴, 𝐵) = ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wo 709   = wceq 1364  wcel 2160  {cab 2175  {crab 2472  cun 3142  ifcif 3549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-un 3148  df-if 3550
This theorem is referenced by:  ifeq1  3552  ifeq2  3553  dfif3  3562  ifssun  3563
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