Theorem dfif3 3673

Description: Alternate definition of the conditional operator df-if 3664. Note that φ is independent of x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)

Hypothesis

Ref	Expression
dfif3.1	⊢ C = {x ∣ φ}

Assertion

Ref	Expression
dfif3	⊢ if(φ, A, B) = ((A ∩ C) ∪ (B ∩ (V ∖ C)))

Distinct variable group:   φ,x

Allowed substitution hints:   A(x)   B(x)   C(x)

Proof of Theorem dfif3

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfif6 3666	. 2 ⊢ if(φ, A, B) = ({y ∈ A ∣ φ} ∪ {y ∈ B ∣ ¬ φ})
2		dfif3.1	. . . . . 6 ⊢ C = {x ∣ φ}
3		biidd 228	. . . . . . 7 ⊢ (x = y → (φ ↔ φ))
4	3	cbvabv 2473	. . . . . 6 ⊢ {x ∣ φ} = {y ∣ φ}
5	2, 4	eqtri 2373	. . . . 5 ⊢ C = {y ∣ φ}
6	5	ineq2i 3455	. . . 4 ⊢ (A ∩ C) = (A ∩ {y ∣ φ})
7		dfrab3 3532	. . . 4 ⊢ {y ∈ A ∣ φ} = (A ∩ {y ∣ φ})
8	6, 7	eqtr4i 2376	. . 3 ⊢ (A ∩ C) = {y ∈ A ∣ φ}
9		dfrab3 3532	. . . 4 ⊢ {y ∈ B ∣ ¬ φ} = (B ∩ {y ∣ ¬ φ})
10		notab 3526	. . . . . 6 ⊢ {y ∣ ¬ φ} = (V ∖ {y ∣ φ})
11	5	difeq2i 3383	. . . . . 6 ⊢ (V ∖ C) = (V ∖ {y ∣ φ})
12	10, 11	eqtr4i 2376	. . . . 5 ⊢ {y ∣ ¬ φ} = (V ∖ C)
13	12	ineq2i 3455	. . . 4 ⊢ (B ∩ {y ∣ ¬ φ}) = (B ∩ (V ∖ C))
14	9, 13	eqtr2i 2374	. . 3 ⊢ (B ∩ (V ∖ C)) = {y ∈ B ∣ ¬ φ}
15	8, 14	uneq12i 3417	. 2 ⊢ ((A ∩ C) ∪ (B ∩ (V ∖ C))) = ({y ∈ A ∣ φ} ∪ {y ∈ B ∣ ¬ φ})
16	1, 15	eqtr4i 2376	1 ⊢ if(φ, A, B) = ((A ∩ C) ∪ (B ∩ (V ∖ C)))

Syntax hints:  ¬ wn 3   = wceq 1642  {cab 2339  {crab 2619  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209   ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-if 3664

This theorem is referenced by:  dfif4  3674