Theorem dfrab2 3500

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 2531	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		inab 3493	. . 3 |- ( { x | x e. A } i^i { x | ph } ) = { x | ( x e. A /\ ph ) }
3		abid2 2357	. . . 4 |- { x | x e. A } = A
4	3	ineq1i 3422	. . 3 |- ( { x | x e. A } i^i { x | ph } ) = ( A i^i { x | ph } )
5	2, 4	eqtr3i 2257	. 2 |- { x | ( x e. A /\ ph ) } = ( A i^i { x | ph } )
6		incom 3415	. 2 |- ( A i^i { x | ph } ) = ( { x | ph } i^i A )
7	1, 5, 6	3eqtri 2259	1 |- { x e. A | ph } = ( { x | ph } i^i A )

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2205   {cab 2220   {crab 2526   i^i cin 3213

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531  df-v 2817  df-in 3220

This theorem is referenced by:  minmax  11940  xrminmax  11975