Theorem dfif5 3674

Description: Alternate definition of the conditional operator df-if 3663. Note that ph is independent of x i.e. a constant true or false (see also abvor0 3567). (Contributed by Gérard Lang, 18-Aug-2013.)

Hypothesis

Ref	Expression
dfif3.1	|- C = {x | ph}

Assertion

Ref	Expression
dfif5	|- if(ph, A, B) = ((A i^i B) u. (((A \ B) i^i C) u. ((B \ A) i^i (_V \ C))))

Distinct variable group:   ph,x

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

Proof of Theorem dfif5

Step	Hyp	Ref	Expression
1		inindi 3472	. 2 |- ((A u. B) i^i ((A u. (_V \ C)) i^i (B u. C))) = (((A u. B) i^i (A u. (_V \ C))) i^i ((A u. B) i^i (B u. C)))
2		dfif3.1	. . 3 |- C = {x | ph}
3	2	dfif4 3673	. 2 |- if(ph, A, B) = ((A u. B) i^i ((A u. (_V \ C)) i^i (B u. C)))
4		undir 3504	. . 3 |- ((A i^i B) u. (((A \ B) i^i C) u. ((B \ A) i^i (_V \ C)))) = ((A u. (((A \ B) i^i C) u. ((B \ A) i^i (_V \ C)))) i^i (B u. (((A \ B) i^i C) u. ((B \ A) i^i (_V \ C)))))
5		unidm 3407	. . . . . . . 8 |- (A u. A) = A
6	5	uneq1i 3414	. . . . . . 7 |- ((A u. A) u. (B i^i (_V \ C))) = (A u. (B i^i (_V \ C)))
7		unass 3420	. . . . . . 7 |- ((A u. A) u. (B i^i (_V \ C))) = (A u. (A u. (B i^i (_V \ C))))
8		undi 3502	. . . . . . 7 |- (A u. (B i^i (_V \ C))) = ((A u. B) i^i (A u. (_V \ C)))
9	6, 7, 8	3eqtr3ri 2382	. . . . . 6 |- ((A u. B) i^i (A u. (_V \ C))) = (A u. (A u. (B i^i (_V \ C))))
10		undi 3502	. . . . . . . 8 |- (A u. ((A \ B) i^i C)) = ((A u. (A \ B)) i^i (A u. C))
11		undifabs 3627	. . . . . . . . . 10 |- (A u. (A \ B)) = A
12	11	ineq1i 3453	. . . . . . . . 9 |- ((A u. (A \ B)) i^i (A u. C)) = (A i^i (A u. C))
13		inabs 3486	. . . . . . . . 9 |- (A i^i (A u. C)) = A
14	12, 13	eqtri 2373	. . . . . . . 8 |- ((A u. (A \ B)) i^i (A u. C)) = A
15	10, 14	eqtri 2373	. . . . . . 7 |- (A u. ((A \ B) i^i C)) = A
16		undif2 3626	. . . . . . . . 9 |- (A u. (B \ A)) = (A u. B)
17	16	ineq1i 3453	. . . . . . . 8 |- ((A u. (B \ A)) i^i (A u. (_V \ C))) = ((A u. B) i^i (A u. (_V \ C)))
18		undi 3502	. . . . . . . 8 |- (A u. ((B \ A) i^i (_V \ C))) = ((A u. (B \ A)) i^i (A u. (_V \ C)))
19	17, 18, 8	3eqtr4i 2383	. . . . . . 7 |- (A u. ((B \ A) i^i (_V \ C))) = (A u. (B i^i (_V \ C)))
20	15, 19	uneq12i 3416	. . . . . 6 |- ((A u. ((A \ B) i^i C)) u. (A u. ((B \ A) i^i (_V \ C)))) = (A u. (A u. (B i^i (_V \ C))))
21	9, 20	eqtr4i 2376	. . . . 5 |- ((A u. B) i^i (A u. (_V \ C))) = ((A u. ((A \ B) i^i C)) u. (A u. ((B \ A) i^i (_V \ C))))
22		unundi 3424	. . . . 5 |- (A u. (((A \ B) i^i C) u. ((B \ A) i^i (_V \ C)))) = ((A u. ((A \ B) i^i C)) u. (A u. ((B \ A) i^i (_V \ C))))
23	21, 22	eqtr4i 2376	. . . 4 |- ((A u. B) i^i (A u. (_V \ C))) = (A u. (((A \ B) i^i C) u. ((B \ A) i^i (_V \ C))))
24		unass 3420	. . . . . 6 |- (((A i^i C) u. B) u. B) = ((A i^i C) u. (B u. B))
25		undi 3502	. . . . . . . . 9 |- (B u. (A i^i C)) = ((B u. A) i^i (B u. C))
26		uncom 3408	. . . . . . . . 9 |- ((A i^i C) u. B) = (B u. (A i^i C))
27		undif2 3626	. . . . . . . . . 10 |- (B u. (A \ B)) = (B u. A)
28	27	ineq1i 3453	. . . . . . . . 9 |- ((B u. (A \ B)) i^i (B u. C)) = ((B u. A) i^i (B u. C))
29	25, 26, 28	3eqtr4i 2383	. . . . . . . 8 |- ((A i^i C) u. B) = ((B u. (A \ B)) i^i (B u. C))
30		undi 3502	. . . . . . . 8 |- (B u. ((A \ B) i^i C)) = ((B u. (A \ B)) i^i (B u. C))
31	29, 30	eqtr4i 2376	. . . . . . 7 |- ((A i^i C) u. B) = (B u. ((A \ B) i^i C))
32		undi 3502	. . . . . . . 8 |- (B u. ((B \ A) i^i (_V \ C))) = ((B u. (B \ A)) i^i (B u. (_V \ C)))
33		undifabs 3627	. . . . . . . . 9 |- (B u. (B \ A)) = B
34	33	ineq1i 3453	. . . . . . . 8 |- ((B u. (B \ A)) i^i (B u. (_V \ C))) = (B i^i (B u. (_V \ C)))
35		inabs 3486	. . . . . . . 8 |- (B i^i (B u. (_V \ C))) = B
36	32, 34, 35	3eqtrri 2378	. . . . . . 7 |- B = (B u. ((B \ A) i^i (_V \ C)))
37	31, 36	uneq12i 3416	. . . . . 6 |- (((A i^i C) u. B) u. B) = ((B u. ((A \ B) i^i C)) u. (B u. ((B \ A) i^i (_V \ C))))
38		unidm 3407	. . . . . . 7 |- (B u. B) = B
39	38	uneq2i 3415	. . . . . 6 |- ((A i^i C) u. (B u. B)) = ((A i^i C) u. B)
40	24, 37, 39	3eqtr3ri 2382	. . . . 5 |- ((A i^i C) u. B) = ((B u. ((A \ B) i^i C)) u. (B u. ((B \ A) i^i (_V \ C))))
41		uncom 3408	. . . . . . 7 |- (B u. C) = (C u. B)
42	41	ineq2i 3454	. . . . . 6 |- ((A u. B) i^i (B u. C)) = ((A u. B) i^i (C u. B))
43		undir 3504	. . . . . 6 |- ((A i^i C) u. B) = ((A u. B) i^i (C u. B))
44	42, 43	eqtr4i 2376	. . . . 5 |- ((A u. B) i^i (B u. C)) = ((A i^i C) u. B)
45		unundi 3424	. . . . 5 |- (B u. (((A \ B) i^i C) u. ((B \ A) i^i (_V \ C)))) = ((B u. ((A \ B) i^i C)) u. (B u. ((B \ A) i^i (_V \ C))))
46	40, 44, 45	3eqtr4i 2383	. . . 4 |- ((A u. B) i^i (B u. C)) = (B u. (((A \ B) i^i C) u. ((B \ A) i^i (_V \ C))))
47	23, 46	ineq12i 3455	. . 3 |- (((A u. B) i^i (A u. (_V \ C))) i^i ((A u. B) i^i (B u. C))) = ((A u. (((A \ B) i^i C) u. ((B \ A) i^i (_V \ C)))) i^i (B u. (((A \ B) i^i C) u. ((B \ A) i^i (_V \ C)))))
48	4, 47	eqtr4i 2376	. 2 |- ((A i^i B) u. (((A \ B) i^i C) u. ((B \ A) i^i (_V \ C)))) = (((A u. B) i^i (A u. (_V \ C))) i^i ((A u. B) i^i (B u. C)))
49	1, 3, 48	3eqtr4i 2383	1 |- if(ph, A, B) = ((A i^i B) u. (((A \ B) i^i C) u. ((B \ A) i^i (_V \ C))))

Syntax hints:   = wceq 1642  {cab 2339  _Vcvv 2859   \ cdif 3206   u. cun 3207   i^i cin 3208  ifcif 3662

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 2478  df-ne 2518  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-if 3663

This theorem is referenced by: (None)