Theorem dfrab3 3480

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 2517	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		inab 3472	. 2 |- ( { x | x e. A } i^i { x | ph } ) = { x | ( x e. A /\ ph ) }
3		abid2 2350	. . 3 |- { x | x e. A } = A
4	3	ineq1i 3401	. 2 |- ( { x | x e. A } i^i { x | ph } ) = ( A i^i { x | ph } )
5	1, 2, 4	3eqtr2i 2256	1 |- { x e. A | ph } = ( A i^i { x | ph } )

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   {cab 2215   {crab 2512   i^i cin 3196

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-in 3203

This theorem is referenced by:  notrab  3481  dfrab3ss  3482  dfif3  3616  dfse2  5101