Theorem dfif3 3570

Description: Alternate definition of the conditional operator df-if 3558. 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 3559	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑})
2		dfif3.1	. . . . . 6 ⊢ 𝐶 = {𝑥 ∣ 𝜑}
3		biidd 172	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
4	3	cbvabv 2318	. . . . . 6 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜑}
5	2, 4	eqtri 2214	. . . . 5 ⊢ 𝐶 = {𝑦 ∣ 𝜑}
6	5	ineq2i 3357	. . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∩ {𝑦 ∣ 𝜑})
7		dfrab3 3435	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑦 ∣ 𝜑})
8	6, 7	eqtr4i 2217	. . 3 ⊢ (𝐴 ∩ 𝐶) = {𝑦 ∈ 𝐴 ∣ 𝜑}
9		dfrab3 3435	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑} = (𝐵 ∩ {𝑦 ∣ ¬ 𝜑})
10		notab 3429	. . . . . 6 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜑})
11	5	difeq2i 3274	. . . . . 6 ⊢ (V ∖ 𝐶) = (V ∖ {𝑦 ∣ 𝜑})
12	10, 11	eqtr4i 2217	. . . . 5 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ 𝐶)
13	12	ineq2i 3357	. . . 4 ⊢ (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) = (𝐵 ∩ (V ∖ 𝐶))
14	9, 13	eqtr2i 2215	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = {𝑦 ∈ 𝐵 ∣ ¬ 𝜑}
15	8, 14	uneq12i 3311	. 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑})
16	1, 15	eqtr4i 2217	1 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))

Syntax hints:  ¬ wn 3   = wceq 1364  {cab 2179  {crab 2476  Vcvv 2760   ∖ cdif 3150   ∪ cun 3151   ∩ cin 3152  ifcif 3557

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-if 3558

This theorem is referenced by: (None)