Theorem dfif5 3675

Description: Alternate definition of the conditional operator df-if 3664. Note that φ is independent of x i.e. a constant true or false (see also abvor0 3568). (Contributed by Gérard Lang, 18-Aug-2013.)

Hypothesis

Ref	Expression
dfif3.1	⊢ C = {x ∣ φ}

Assertion

Ref	Expression
dfif5	⊢ if(φ, A, B) = ((A ∩ B) ∪ (((A ∖ B) ∩ C) ∪ ((B ∖ A) ∩ (V ∖ C))))

Distinct variable group:   φ,x

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

Proof of Theorem dfif5

Step	Hyp	Ref	Expression
1		inindi 3473	. 2 ⊢ ((A ∪ B) ∩ ((A ∪ (V ∖ C)) ∩ (B ∪ C))) = (((A ∪ B) ∩ (A ∪ (V ∖ C))) ∩ ((A ∪ B) ∩ (B ∪ C)))
2		dfif3.1	. . 3 ⊢ C = {x ∣ φ}
3	2	dfif4 3674	. 2 ⊢ if(φ, A, B) = ((A ∪ B) ∩ ((A ∪ (V ∖ C)) ∩ (B ∪ C)))
4		undir 3505	. . 3 ⊢ ((A ∩ B) ∪ (((A ∖ B) ∩ C) ∪ ((B ∖ A) ∩ (V ∖ C)))) = ((A ∪ (((A ∖ B) ∩ C) ∪ ((B ∖ A) ∩ (V ∖ C)))) ∩ (B ∪ (((A ∖ B) ∩ C) ∪ ((B ∖ A) ∩ (V ∖ C)))))
5		unidm 3408	. . . . . . . 8 ⊢ (A ∪ A) = A
6	5	uneq1i 3415	. . . . . . 7 ⊢ ((A ∪ A) ∪ (B ∩ (V ∖ C))) = (A ∪ (B ∩ (V ∖ C)))
7		unass 3421	. . . . . . 7 ⊢ ((A ∪ A) ∪ (B ∩ (V ∖ C))) = (A ∪ (A ∪ (B ∩ (V ∖ C))))
8		undi 3503	. . . . . . 7 ⊢ (A ∪ (B ∩ (V ∖ C))) = ((A ∪ B) ∩ (A ∪ (V ∖ C)))
9	6, 7, 8	3eqtr3ri 2382	. . . . . 6 ⊢ ((A ∪ B) ∩ (A ∪ (V ∖ C))) = (A ∪ (A ∪ (B ∩ (V ∖ C))))
10		undi 3503	. . . . . . . 8 ⊢ (A ∪ ((A ∖ B) ∩ C)) = ((A ∪ (A ∖ B)) ∩ (A ∪ C))
11		undifabs 3628	. . . . . . . . . 10 ⊢ (A ∪ (A ∖ B)) = A
12	11	ineq1i 3454	. . . . . . . . 9 ⊢ ((A ∪ (A ∖ B)) ∩ (A ∪ C)) = (A ∩ (A ∪ C))
13		inabs 3487	. . . . . . . . 9 ⊢ (A ∩ (A ∪ C)) = A
14	12, 13	eqtri 2373	. . . . . . . 8 ⊢ ((A ∪ (A ∖ B)) ∩ (A ∪ C)) = A
15	10, 14	eqtri 2373	. . . . . . 7 ⊢ (A ∪ ((A ∖ B) ∩ C)) = A
16		undif2 3627	. . . . . . . . 9 ⊢ (A ∪ (B ∖ A)) = (A ∪ B)
17	16	ineq1i 3454	. . . . . . . 8 ⊢ ((A ∪ (B ∖ A)) ∩ (A ∪ (V ∖ C))) = ((A ∪ B) ∩ (A ∪ (V ∖ C)))
18		undi 3503	. . . . . . . 8 ⊢ (A ∪ ((B ∖ A) ∩ (V ∖ C))) = ((A ∪ (B ∖ A)) ∩ (A ∪ (V ∖ C)))
19	17, 18, 8	3eqtr4i 2383	. . . . . . 7 ⊢ (A ∪ ((B ∖ A) ∩ (V ∖ C))) = (A ∪ (B ∩ (V ∖ C)))
20	15, 19	uneq12i 3417	. . . . . 6 ⊢ ((A ∪ ((A ∖ B) ∩ C)) ∪ (A ∪ ((B ∖ A) ∩ (V ∖ C)))) = (A ∪ (A ∪ (B ∩ (V ∖ C))))
21	9, 20	eqtr4i 2376	. . . . 5 ⊢ ((A ∪ B) ∩ (A ∪ (V ∖ C))) = ((A ∪ ((A ∖ B) ∩ C)) ∪ (A ∪ ((B ∖ A) ∩ (V ∖ C))))
22		unundi 3425	. . . . 5 ⊢ (A ∪ (((A ∖ B) ∩ C) ∪ ((B ∖ A) ∩ (V ∖ C)))) = ((A ∪ ((A ∖ B) ∩ C)) ∪ (A ∪ ((B ∖ A) ∩ (V ∖ C))))
23	21, 22	eqtr4i 2376	. . . 4 ⊢ ((A ∪ B) ∩ (A ∪ (V ∖ C))) = (A ∪ (((A ∖ B) ∩ C) ∪ ((B ∖ A) ∩ (V ∖ C))))
24		unass 3421	. . . . . 6 ⊢ (((A ∩ C) ∪ B) ∪ B) = ((A ∩ C) ∪ (B ∪ B))
25		undi 3503	. . . . . . . . 9 ⊢ (B ∪ (A ∩ C)) = ((B ∪ A) ∩ (B ∪ C))
26		uncom 3409	. . . . . . . . 9 ⊢ ((A ∩ C) ∪ B) = (B ∪ (A ∩ C))
27		undif2 3627	. . . . . . . . . 10 ⊢ (B ∪ (A ∖ B)) = (B ∪ A)
28	27	ineq1i 3454	. . . . . . . . 9 ⊢ ((B ∪ (A ∖ B)) ∩ (B ∪ C)) = ((B ∪ A) ∩ (B ∪ C))
29	25, 26, 28	3eqtr4i 2383	. . . . . . . 8 ⊢ ((A ∩ C) ∪ B) = ((B ∪ (A ∖ B)) ∩ (B ∪ C))
30		undi 3503	. . . . . . . 8 ⊢ (B ∪ ((A ∖ B) ∩ C)) = ((B ∪ (A ∖ B)) ∩ (B ∪ C))
31	29, 30	eqtr4i 2376	. . . . . . 7 ⊢ ((A ∩ C) ∪ B) = (B ∪ ((A ∖ B) ∩ C))
32		undi 3503	. . . . . . . 8 ⊢ (B ∪ ((B ∖ A) ∩ (V ∖ C))) = ((B ∪ (B ∖ A)) ∩ (B ∪ (V ∖ C)))
33		undifabs 3628	. . . . . . . . 9 ⊢ (B ∪ (B ∖ A)) = B
34	33	ineq1i 3454	. . . . . . . 8 ⊢ ((B ∪ (B ∖ A)) ∩ (B ∪ (V ∖ C))) = (B ∩ (B ∪ (V ∖ C)))
35		inabs 3487	. . . . . . . 8 ⊢ (B ∩ (B ∪ (V ∖ C))) = B
36	32, 34, 35	3eqtrri 2378	. . . . . . 7 ⊢ B = (B ∪ ((B ∖ A) ∩ (V ∖ C)))
37	31, 36	uneq12i 3417	. . . . . 6 ⊢ (((A ∩ C) ∪ B) ∪ B) = ((B ∪ ((A ∖ B) ∩ C)) ∪ (B ∪ ((B ∖ A) ∩ (V ∖ C))))
38		unidm 3408	. . . . . . 7 ⊢ (B ∪ B) = B
39	38	uneq2i 3416	. . . . . 6 ⊢ ((A ∩ C) ∪ (B ∪ B)) = ((A ∩ C) ∪ B)
40	24, 37, 39	3eqtr3ri 2382	. . . . 5 ⊢ ((A ∩ C) ∪ B) = ((B ∪ ((A ∖ B) ∩ C)) ∪ (B ∪ ((B ∖ A) ∩ (V ∖ C))))
41		uncom 3409	. . . . . . 7 ⊢ (B ∪ C) = (C ∪ B)
42	41	ineq2i 3455	. . . . . 6 ⊢ ((A ∪ B) ∩ (B ∪ C)) = ((A ∪ B) ∩ (C ∪ B))
43		undir 3505	. . . . . 6 ⊢ ((A ∩ C) ∪ B) = ((A ∪ B) ∩ (C ∪ B))
44	42, 43	eqtr4i 2376	. . . . 5 ⊢ ((A ∪ B) ∩ (B ∪ C)) = ((A ∩ C) ∪ B)
45		unundi 3425	. . . . 5 ⊢ (B ∪ (((A ∖ B) ∩ C) ∪ ((B ∖ A) ∩ (V ∖ C)))) = ((B ∪ ((A ∖ B) ∩ C)) ∪ (B ∪ ((B ∖ A) ∩ (V ∖ C))))
46	40, 44, 45	3eqtr4i 2383	. . . 4 ⊢ ((A ∪ B) ∩ (B ∪ C)) = (B ∪ (((A ∖ B) ∩ C) ∪ ((B ∖ A) ∩ (V ∖ C))))
47	23, 46	ineq12i 3456	. . 3 ⊢ (((A ∪ B) ∩ (A ∪ (V ∖ C))) ∩ ((A ∪ B) ∩ (B ∪ C))) = ((A ∪ (((A ∖ B) ∩ C) ∪ ((B ∖ A) ∩ (V ∖ C)))) ∩ (B ∪ (((A ∖ B) ∩ C) ∪ ((B ∖ A) ∩ (V ∖ C)))))
48	4, 47	eqtr4i 2376	. 2 ⊢ ((A ∩ B) ∪ (((A ∖ B) ∩ C) ∪ ((B ∖ A) ∩ (V ∖ C)))) = (((A ∪ B) ∩ (A ∪ (V ∖ C))) ∩ ((A ∪ B) ∩ (B ∪ C)))
49	1, 3, 48	3eqtr4i 2383	1 ⊢ if(φ, A, B) = ((A ∩ B) ∪ (((A ∖ B) ∩ C) ∪ ((B ∖ A) ∩ (V ∖ C))))

Syntax hints:   = wceq 1642  {cab 2339  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209   ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-if 3664

This theorem is referenced by: (None)