Theorem dfif6 3637

Description: An alternate definition of the conditional operator df-if 3636 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
dfif6	⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfif6

Step	Hyp	Ref	Expression
1		unab 3498	. 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)}) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
2		df-rab 2537	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3		df-rab 2537	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)}
4	2, 3	uneq12i 3381	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)})
5		df-if 3636	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
6	1, 4, 5	3eqtr4ri 2270	1 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 720   = wceq 1402   ∈ wcel 2209  {cab 2224  {crab 2532   ∪ cun 3218  ifcif 3635

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-un 3224  df-if 3636

This theorem is referenced by:  if0ab  3638  ifeq1  3640  ifeq2  3641  dfif3  3651  ifssun  3652