Theorem dfif6 3415

Description: An alternate definition of the conditional operator df-if 3414 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 3282	. 2 |- ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ -. ph ) } ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
2		df-rab 2379	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
3		df-rab 2379	. . 3 |- { x e. B | -. ph } = { x | ( x e. B /\ -. ph ) }
4	2, 3	uneq12i 3167	. 2 |- ( { x e. A | ph } u. { x e. B | -. ph } ) = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ -. ph ) } )
5		df-if 3414	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
6	1, 4, 5	3eqtr4ri 2126	1 |- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )

Syntax hints:   -. wn 3   /\ wa 103   \/ wo 667   = wceq 1296   e. wcel 1445   {cab 2081   {crab 2374   u. cun 3011   ifcif 3413

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rab 2379  df-v 2635  df-un 3017  df-if 3414

This theorem is referenced by:  ifeq1  3416  ifeq2  3417  dfif3  3426