Theorem dfrab2 3434

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 2481	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		inab 3427	. . 3 |- ( { x | x e. A } i^i { x | ph } ) = { x | ( x e. A /\ ph ) }
3		abid2 2314	. . . 4 |- { x | x e. A } = A
4	3	ineq1i 3356	. . 3 |- ( { x | x e. A } i^i { x | ph } ) = ( A i^i { x | ph } )
5	2, 4	eqtr3i 2216	. 2 |- { x | ( x e. A /\ ph ) } = ( A i^i { x | ph } )
6		incom 3351	. 2 |- ( A i^i { x | ph } ) = ( { x | ph } i^i A )
7	1, 5, 6	3eqtri 2218	1 |- { x e. A | ph } = ( { x | ph } i^i A )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2164   {cab 2179   {crab 2476   i^i cin 3152

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-in 3159

This theorem is referenced by:  minmax  11373  xrminmax  11408