Theorem dfrab3 3409

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 2462	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		inab 3401	. 2 |- ( { x | x e. A } i^i { x | ph } ) = { x | ( x e. A /\ ph ) }
3		abid2 2296	. . 3 |- { x | x e. A } = A
4	3	ineq1i 3330	. 2 |- ( { x | x e. A } i^i { x | ph } ) = ( A i^i { x | ph } )
5	1, 2, 4	3eqtr2i 2202	1 |- { x e. A | ph } = ( A i^i { x | ph } )

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2146   {cab 2161   {crab 2457   i^i cin 3126

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rab 2462  df-v 2737  df-in 3133

This theorem is referenced by:  notrab  3410  dfrab3ss  3411  dfif3  3545  dfse2  4994