Theorem dfif6 3607

Description: An alternate definition of the conditional operator df-if 3606 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 3474	. 2 |- ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ -. ph ) } ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
2		df-rab 2519	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
3		df-rab 2519	. . 3 |- { x e. B | -. ph } = { x | ( x e. B /\ -. ph ) }
4	2, 3	uneq12i 3359	. 2 |- ( { x e. A | ph } u. { x e. B | -. ph } ) = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ -. ph ) } )
5		df-if 3606	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
6	1, 4, 5	3eqtr4ri 2263	1 |- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 715   = wceq 1397   e. wcel 2202   {cab 2217   {crab 2514   u. cun 3198   ifcif 3605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-un 3204  df-if 3606

This theorem is referenced by:  ifeq1  3608  ifeq2  3609  dfif3  3619  ifssun  3620