Theorem dfif3 3539

Description: Alternate definition of the conditional operator df-if 3527. 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 3528	. 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 2295	. . . . . 6 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜑}
5	2, 4	eqtri 2191	. . . . 5 ⊢ 𝐶 = {𝑦 ∣ 𝜑}
6	5	ineq2i 3325	. . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∩ {𝑦 ∣ 𝜑})
7		dfrab3 3403	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑦 ∣ 𝜑})
8	6, 7	eqtr4i 2194	. . 3 ⊢ (𝐴 ∩ 𝐶) = {𝑦 ∈ 𝐴 ∣ 𝜑}
9		dfrab3 3403	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑} = (𝐵 ∩ {𝑦 ∣ ¬ 𝜑})
10		notab 3397	. . . . . 6 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜑})
11	5	difeq2i 3242	. . . . . 6 ⊢ (V ∖ 𝐶) = (V ∖ {𝑦 ∣ 𝜑})
12	10, 11	eqtr4i 2194	. . . . 5 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ 𝐶)
13	12	ineq2i 3325	. . . 4 ⊢ (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) = (𝐵 ∩ (V ∖ 𝐶))
14	9, 13	eqtr2i 2192	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = {𝑦 ∈ 𝐵 ∣ ¬ 𝜑}
15	8, 14	uneq12i 3279	. 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑})
16	1, 15	eqtr4i 2194	1 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))

Syntax hints:  ¬ wn 3   = wceq 1348  {cab 2156  {crab 2452  Vcvv 2730   ∖ cdif 3118   ∪ cun 3119   ∩ cin 3120  ifcif 3526

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-if 3527

This theorem is referenced by: (None)