Theorem dfif6 3665

Description: An alternate definition of the conditional operator df-if 3663 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 3521	. 2 ⊢ ({x ∣ (x ∈ A ∧ φ)} ∪ {x ∣ (x ∈ B ∧ ¬ φ)}) = {x ∣ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))}
2		df-rab 2623	. . 3 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
3		df-rab 2623	. . 3 ⊢ {x ∈ B ∣ ¬ φ} = {x ∣ (x ∈ B ∧ ¬ φ)}
4	2, 3	uneq12i 3416	. 2 ⊢ ({x ∈ A ∣ φ} ∪ {x ∈ B ∣ ¬ φ}) = ({x ∣ (x ∈ A ∧ φ)} ∪ {x ∣ (x ∈ B ∧ ¬ φ)})
5		df-if 3663	. 2 ⊢ if(φ, A, B) = {x ∣ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))}
6	1, 4, 5	3eqtr4ri 2384	1 ⊢ if(φ, A, B) = ({x ∈ A ∣ φ} ∪ {x ∈ B ∣ ¬ φ})

Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618   ∪ cun 3207   ifcif 3662

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-if 3663

This theorem is referenced by:  ifeq1  3666  ifeq2  3667  dfif3  3672