Theorem rabab 2843

Description: A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Assertion

Ref	Expression
rabab	|- { x e. _V | ph } = { x | ph }

Proof of Theorem rabab

Step	Hyp	Ref	Expression
1		df-rab 2537	. 2 |- { x e. _V | ph } = { x | ( x e. _V /\ ph ) }
2		vex 2824	. . . 4 |- x e. _V
3	2	biantrur 303	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	abbii 2354	. 2 |- { x | ph } = { x | ( x e. _V /\ ph ) }
5	1, 4	eqtr4i 2262	1 |- { x e. _V | ph } = { x | ph }

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   {crab 2532   _Vcvv 2821

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-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  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-rab 2537  df-v 2823

This theorem is referenced by:  notab  3503  intmin2  3991  euen1  7079  bj-omind  16874