Theorem dfif4 4471

Description: Alternate definition of the conditional operator df-if 4457. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)

Hypothesis

Ref	Expression
dfif3.1	⊢ 𝐶 = {𝑥 ∣ 𝜑}

Assertion

Ref	Expression
dfif4	⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶)))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem dfif4

Step	Hyp	Ref	Expression
1		dfif3.1	. . 3 ⊢ 𝐶 = {𝑥 ∣ 𝜑}
2	1	dfif3 4470	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))
3		undir 4207	. 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))))
4		undi 4205	. . . 4 ⊢ (𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶)))
5		undi 4205	. . . . 5 ⊢ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐶 ∪ 𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶)))
6		uncom 4083	. . . . . 6 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶)
7		unvdif 4405	. . . . . 6 ⊢ (𝐶 ∪ (V ∖ 𝐶)) = V
8	6, 7	ineq12i 4141	. . . . 5 ⊢ ((𝐶 ∪ 𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶))) = ((𝐵 ∪ 𝐶) ∩ V)
9		inv1 4325	. . . . 5 ⊢ ((𝐵 ∪ 𝐶) ∩ V) = (𝐵 ∪ 𝐶)
10	5, 8, 9	3eqtri 2770	. . . 4 ⊢ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = (𝐵 ∪ 𝐶)
11	4, 10	ineq12i 4141	. . 3 ⊢ ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = (((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵 ∪ 𝐶))
12		inass 4150	. . 3 ⊢ (((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶)))
13	11, 12	eqtri 2766	. 2 ⊢ ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶)))
14	2, 3, 13	3eqtri 2770	1 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶)))

Syntax hints:   = wceq 1539  {cab 2715  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882  ifcif 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457

This theorem is referenced by:  dfif5  4472  ifssun  4473