Theorem dfrab2 3356

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 2426	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		inab 3349	. . 3 |- ( { x | x e. A } i^i { x | ph } ) = { x | ( x e. A /\ ph ) }
3		abid2 2261	. . . 4 |- { x | x e. A } = A
4	3	ineq1i 3278	. . 3 |- ( { x | x e. A } i^i { x | ph } ) = ( A i^i { x | ph } )
5	2, 4	eqtr3i 2163	. 2 |- { x | ( x e. A /\ ph ) } = ( A i^i { x | ph } )
6		incom 3273	. 2 |- ( A i^i { x | ph } ) = ( { x | ph } i^i A )
7	1, 5, 6	3eqtri 2165	1 |- { x e. A | ph } = ( { x | ph } i^i A )

Syntax hints:   /\ wa 103   = wceq 1332   e. wcel 1481   {cab 2126   {crab 2421   i^i cin 3075

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691  df-in 3082

This theorem is referenced by:  minmax  11033  xrminmax  11066