Theorem dfrab2 3482

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 2519	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		inab 3475	. . 3 |- ( { x | x e. A } i^i { x | ph } ) = { x | ( x e. A /\ ph ) }
3		abid2 2352	. . . 4 |- { x | x e. A } = A
4	3	ineq1i 3404	. . 3 |- ( { x | x e. A } i^i { x | ph } ) = ( A i^i { x | ph } )
5	2, 4	eqtr3i 2254	. 2 |- { x | ( x e. A /\ ph ) } = ( A i^i { x | ph } )
6		incom 3399	. 2 |- ( A i^i { x | ph } ) = ( { x | ph } i^i A )
7	1, 5, 6	3eqtri 2256	1 |- { x e. A | ph } = ( { x | ph } i^i A )

Syntax hints:   /\ wa 104   = wceq 1397   e. wcel 2202   {cab 2217   {crab 2514   i^i cin 3199

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-in 3206

This theorem is referenced by:  minmax  11790  xrminmax  11825