Theorem dfif4 3674

Description: Alternate definition of the conditional operator df-if 3664. 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 3673	. 2 ⊢ if(φ, A, B) = ((A ∩ C) ∪ (B ∩ (V ∖ C)))
3		undir 3505	. 2 ⊢ ((A ∩ C) ∪ (B ∩ (V ∖ C))) = ((A ∪ (B ∩ (V ∖ C))) ∩ (C ∪ (B ∩ (V ∖ C))))
4		undi 3503	. . . 4 ⊢ (A ∪ (B ∩ (V ∖ C))) = ((A ∪ B) ∩ (A ∪ (V ∖ C)))
5		undi 3503	. . . . 5 ⊢ (C ∪ (B ∩ (V ∖ C))) = ((C ∪ B) ∩ (C ∪ (V ∖ C)))
6		uncom 3409	. . . . . 6 ⊢ (C ∪ B) = (B ∪ C)
7		undifv 3625	. . . . . 6 ⊢ (C ∪ (V ∖ C)) = V
8	6, 7	ineq12i 3456	. . . . 5 ⊢ ((C ∪ B) ∩ (C ∪ (V ∖ C))) = ((B ∪ C) ∩ V)
9		inv1 3578	. . . . 5 ⊢ ((B ∪ C) ∩ V) = (B ∪ C)
10	5, 8, 9	3eqtri 2377	. . . 4 ⊢ (C ∪ (B ∩ (V ∖ C))) = (B ∪ C)
11	4, 10	ineq12i 3456	. . 3 ⊢ ((A ∪ (B ∩ (V ∖ C))) ∩ (C ∪ (B ∩ (V ∖ C)))) = (((A ∪ B) ∩ (A ∪ (V ∖ C))) ∩ (B ∪ C))
12		inass 3466	. . 3 ⊢ (((A ∪ B) ∩ (A ∪ (V ∖ C))) ∩ (B ∪ C)) = ((A ∪ B) ∩ ((A ∪ (V ∖ C)) ∩ (B ∪ C)))
13	11, 12	eqtri 2373	. 2 ⊢ ((A ∪ (B ∩ (V ∖ C))) ∩ (C ∪ (B ∩ (V ∖ C)))) = ((A ∪ B) ∩ ((A ∪ (V ∖ C)) ∩ (B ∪ C)))
14	2, 3, 13	3eqtri 2377	1 ⊢ if(φ, A, B) = ((A ∪ B) ∩ ((A ∪ (V ∖ C)) ∩ (B ∪ 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:  dfif5  3675