Theorem dfif6 3353

Description: An alternate definition of the conditional operator df-if 3352 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
dfif6	|- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )

Distinct variable groups:   ph, x   x, A   x, B

Proof of Theorem dfif6

Step	Hyp	Ref	Expression
1		unab 3231	. 2 |- ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ -. ph ) } ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
2		df-rab 2357	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
3		df-rab 2357	. . 3 |- { x e. B | -. ph } = { x | ( x e. B /\ -. ph ) }
4	2, 3	uneq12i 3124	. 2 |- ( { x e. A | ph } u. { x e. B | -. ph } ) = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ -. ph ) } )
5		df-if 3352	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
6	1, 4, 5	3eqtr4ri 2112	1 |- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )

Syntax hints:   -. wn 3   /\ wa 102   \/ wo 661   = wceq 1284   e. wcel 1433   {cab 2067   {crab 2352   u. cun 2971   ifcif 3351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-un 2977  df-if 3352

This theorem is referenced by:  ifeq1  3354  ifeq2  3355  dfif3  3364