Theorem dfif6 3476

Description: An alternate definition of the conditional operator df-if 3475 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 3343	. 2 |- ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ -. ph ) } ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
2		df-rab 2425	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
3		df-rab 2425	. . 3 |- { x e. B | -. ph } = { x | ( x e. B /\ -. ph ) }
4	2, 3	uneq12i 3228	. 2 |- ( { x e. A | ph } u. { x e. B | -. ph } ) = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ -. ph ) } )
5		df-if 3475	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
6	1, 4, 5	3eqtr4ri 2171	1 |- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )

Syntax hints:   -. wn 3   /\ wa 103   \/ wo 697   = wceq 1331   e. wcel 1480   {cab 2125   {crab 2420   u. cun 3069   ifcif 3474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-un 3075  df-if 3475

This theorem is referenced by:  ifeq1  3477  ifeq2  3478  dfif3  3487