Theorem dfif3 4133

Description: Alternate definition of the conditional operator df-if 4120. 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 4122	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑})
2		dfif3.1	. . . . . 6 ⊢ 𝐶 = {𝑥 ∣ 𝜑}
3		biidd 252	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
4	3	cbvabv 2776	. . . . . 6 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜑}
5	2, 4	eqtri 2673	. . . . 5 ⊢ 𝐶 = {𝑦 ∣ 𝜑}
6	5	ineq2i 3844	. . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∩ {𝑦 ∣ 𝜑})
7		dfrab3 3935	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑦 ∣ 𝜑})
8	6, 7	eqtr4i 2676	. . 3 ⊢ (𝐴 ∩ 𝐶) = {𝑦 ∈ 𝐴 ∣ 𝜑}
9		dfrab3 3935	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑} = (𝐵 ∩ {𝑦 ∣ ¬ 𝜑})
10		notab 3930	. . . . . 6 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜑})
11	5	difeq2i 3758	. . . . . 6 ⊢ (V ∖ 𝐶) = (V ∖ {𝑦 ∣ 𝜑})
12	10, 11	eqtr4i 2676	. . . . 5 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ 𝐶)
13	12	ineq2i 3844	. . . 4 ⊢ (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) = (𝐵 ∩ (V ∖ 𝐶))
14	9, 13	eqtr2i 2674	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = {𝑦 ∈ 𝐵 ∣ ¬ 𝜑}
15	8, 14	uneq12i 3798	. 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑})
16	1, 15	eqtr4i 2676	1 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))

Syntax hints:  ¬ wn 3   = wceq 1523  {cab 2637  {crab 2945  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ∩ cin 3606  ifcif 4119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-if 4120

This theorem is referenced by:  dfif4  4134