Theorem dfif3 4473

Description: Alternate definition of the conditional operator df-if 4460. 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 variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfif6 4462	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑})
2		dfif3.1	. . . . . 6 ⊢ 𝐶 = {𝑥 ∣ 𝜑}
3		biidd 261	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
4	3	cbvabv 2811	. . . . . 6 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜑}
5	2, 4	eqtri 2766	. . . . 5 ⊢ 𝐶 = {𝑦 ∣ 𝜑}
6	5	ineq2i 4143	. . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∩ {𝑦 ∣ 𝜑})
7		dfrab3 4243	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑦 ∣ 𝜑})
8	6, 7	eqtr4i 2769	. . 3 ⊢ (𝐴 ∩ 𝐶) = {𝑦 ∈ 𝐴 ∣ 𝜑}
9		dfrab3 4243	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑} = (𝐵 ∩ {𝑦 ∣ ¬ 𝜑})
10		biidd 261	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜑))
11	10	notabw 4237	. . . . . 6 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ {𝑧 ∣ 𝜑})
12		biidd 261	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜑))
13	12	cbvabv 2811	. . . . . . . 8 ⊢ {𝑥 ∣ 𝜑} = {𝑧 ∣ 𝜑}
14	2, 13	eqtri 2766	. . . . . . 7 ⊢ 𝐶 = {𝑧 ∣ 𝜑}
15	14	difeq2i 4054	. . . . . 6 ⊢ (V ∖ 𝐶) = (V ∖ {𝑧 ∣ 𝜑})
16	11, 15	eqtr4i 2769	. . . . 5 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ 𝐶)
17	16	ineq2i 4143	. . . 4 ⊢ (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) = (𝐵 ∩ (V ∖ 𝐶))
18	9, 17	eqtr2i 2767	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = {𝑦 ∈ 𝐵 ∣ ¬ 𝜑}
19	8, 18	uneq12i 4095	. 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑})
20	1, 19	eqtr4i 2769	1 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))

Syntax hints:  ¬ wn 3   = wceq 1539  {cab 2715  {crab 3068  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886  ifcif 4459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-if 4460

This theorem is referenced by:  dfif4  4474