Theorem dfrab3 3509

Description: Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
dfrab3	|- { x e. A | ph } = ( A i^i { x | ph } )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem dfrab3

Step	Hyp	Ref	Expression
1		df-rab 2537	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		inab 3499	. 2 |- ( { x | x e. A } i^i { x | ph } ) = { x | ( x e. A /\ ph ) }
3		abid2 2361	. . 3 |- { x | x e. A } = A
4	3	ineq1i 3428	. 2 |- ( { x | x e. A } i^i { x | ph } ) = ( A i^i { x | ph } )
5	1, 2, 4	3eqtr2i 2265	1 |- { x e. A | ph } = ( A i^i { x | ph } )

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   {crab 2532   i^i cin 3219

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-in 3226

This theorem is referenced by:  notrab  3510  dfrab3ss  3511  dfif3  3651  dfse2  5155