Theorem dfif4 4506

Description: Alternate definition of the conditional operator df-if 4491. 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 4505	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))
3		undir 4243	. 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))))
4		undi 4241	. . . 4 ⊢ (𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶)))
5		undi 4241	. . . . 5 ⊢ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐶 ∪ 𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶)))
6		uncom 4115	. . . . . 6 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶)
7		unvdif 4439	. . . . . 6 ⊢ (𝐶 ∪ (V ∖ 𝐶)) = V
8	6, 7	ineq12i 4174	. . . . 5 ⊢ ((𝐶 ∪ 𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶))) = ((𝐵 ∪ 𝐶) ∩ V)
9		inv1 4358	. . . . 5 ⊢ ((𝐵 ∪ 𝐶) ∩ V) = (𝐵 ∪ 𝐶)
10	5, 8, 9	3eqtri 2793	. . . 4 ⊢ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = (𝐵 ∪ 𝐶)
11	4, 10	ineq12i 4174	. . 3 ⊢ ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = (((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵 ∪ 𝐶))
12		inass 4183	. . 3 ⊢ (((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶)))
13	11, 12	eqtri 2789	. 2 ⊢ ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶)))
14	2, 3, 13	3eqtri 2793	1 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶)))

This proof depends on syntax axioms:   = wceq 1570  {cab 2744  Vcvv 3458   ∖ cdif 3905   ∪ cun 3906   ∩ cin 3907  ifcif 4490

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4491

This theorem is used by:  dfif5  4507  ifssun  4508