Theorem dfif4 3674

Description: Alternate definition of the conditional operator df-if 3664. Note that ph is independent of x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)

Hypothesis

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

Assertion

Ref	Expression
dfif4	|- if(ph, A, B) = ((A u. B) i^i ((A u. (_V \ C)) i^i (B u. C)))

Distinct variable group:   ph,x

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

Proof of Theorem dfif4

Step	Hyp	Ref	Expression
1		dfif3.1	. . 3 |- C = {x | ph}
2	1	dfif3 3673	. 2 |- if(ph, A, B) = ((A i^i C) u. (B i^i (_V \ C)))
3		undir 3505	. 2 |- ((A i^i C) u. (B i^i (_V \ C))) = ((A u. (B i^i (_V \ C))) i^i (C u. (B i^i (_V \ C))))
4		undi 3503	. . . 4 |- (A u. (B i^i (_V \ C))) = ((A u. B) i^i (A u. (_V \ C)))
5		undi 3503	. . . . 5 |- (C u. (B i^i (_V \ C))) = ((C u. B) i^i (C u. (_V \ C)))
6		uncom 3409	. . . . . 6 |- (C u. B) = (B u. C)
7		undifv 3625	. . . . . 6 |- (C u. (_V \ C)) = _V
8	6, 7	ineq12i 3456	. . . . 5 |- ((C u. B) i^i (C u. (_V \ C))) = ((B u. C) i^i _V)
9		inv1 3578	. . . . 5 |- ((B u. C) i^i _V) = (B u. C)
10	5, 8, 9	3eqtri 2377	. . . 4 |- (C u. (B i^i (_V \ C))) = (B u. C)
11	4, 10	ineq12i 3456	. . 3 |- ((A u. (B i^i (_V \ C))) i^i (C u. (B i^i (_V \ C)))) = (((A u. B) i^i (A u. (_V \ C))) i^i (B u. C))
12		inass 3466	. . 3 |- (((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 (B u. C)))
13	11, 12	eqtri 2373	. 2 |- ((A u. (B i^i (_V \ C))) i^i (C u. (B i^i (_V \ C)))) = ((A u. B) i^i ((A u. (_V \ C)) i^i (B u. C)))
14	2, 3, 13	3eqtri 2377	1 |- if(ph, A, B) = ((A u. B) i^i ((A u. (_V \ C)) i^i (B u. C)))

Syntax hints:   = wceq 1642  {cab 2339  _Vcvv 2860   \ cdif 3207   u. cun 3208   i^i 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