Theorem dfif3 3492

Description: Alternate definition of the conditional operator df-if 3480. 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 3481	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑})
2		dfif3.1	. . . . . 6 ⊢ 𝐶 = {𝑥 ∣ 𝜑}
3		biidd 171	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
4	3	cbvabv 2265	. . . . . 6 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜑}
5	2, 4	eqtri 2161	. . . . 5 ⊢ 𝐶 = {𝑦 ∣ 𝜑}
6	5	ineq2i 3279	. . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∩ {𝑦 ∣ 𝜑})
7		dfrab3 3357	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑦 ∣ 𝜑})
8	6, 7	eqtr4i 2164	. . 3 ⊢ (𝐴 ∩ 𝐶) = {𝑦 ∈ 𝐴 ∣ 𝜑}
9		dfrab3 3357	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑} = (𝐵 ∩ {𝑦 ∣ ¬ 𝜑})
10		notab 3351	. . . . . 6 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜑})
11	5	difeq2i 3196	. . . . . 6 ⊢ (V ∖ 𝐶) = (V ∖ {𝑦 ∣ 𝜑})
12	10, 11	eqtr4i 2164	. . . . 5 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ 𝐶)
13	12	ineq2i 3279	. . . 4 ⊢ (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) = (𝐵 ∩ (V ∖ 𝐶))
14	9, 13	eqtr2i 2162	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = {𝑦 ∈ 𝐵 ∣ ¬ 𝜑}
15	8, 14	uneq12i 3233	. 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑})
16	1, 15	eqtr4i 2164	1 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))

Syntax hints:  ¬ wn 3   = wceq 1332  {cab 2126  {crab 2421  Vcvv 2689   ∖ cdif 3073   ∪ cun 3074   ∩ cin 3075  ifcif 3479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-if 3480

This theorem is referenced by: (None)