Theorem dfrab2 3397

Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)

Assertion

Ref	Expression
dfrab2	|- { x e. A | ph } = ( { x | ph } i^i A )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem dfrab2

Step	Hyp	Ref	Expression
1		df-rab 2453	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		inab 3390	. . 3 |- ( { x | x e. A } i^i { x | ph } ) = { x | ( x e. A /\ ph ) }
3		abid2 2287	. . . 4 |- { x | x e. A } = A
4	3	ineq1i 3319	. . 3 |- ( { x | x e. A } i^i { x | ph } ) = ( A i^i { x | ph } )
5	2, 4	eqtr3i 2188	. 2 |- { x | ( x e. A /\ ph ) } = ( A i^i { x | ph } )
6		incom 3314	. 2 |- ( A i^i { x | ph } ) = ( { x | ph } i^i A )
7	1, 5, 6	3eqtri 2190	1 |- { x e. A | ph } = ( { x | ph } i^i A )

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   {cab 2151   {crab 2448   i^i cin 3115

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-in 3122

This theorem is referenced by:  minmax  11171  xrminmax  11206