Theorem rabab 2747

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 2453	. 2 |- { x e. _V | ph } = { x | ( x e. _V /\ ph ) }
2		vex 2729	. . . 4 |- x e. _V
3	2	biantrur 301	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	abbii 2282	. 2 |- { x | ph } = { x | ( x e. _V /\ ph ) }
5	1, 4	eqtr4i 2189	1 |- { x e. _V | ph } = { x | ph }

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   {cab 2151   {crab 2448   _Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-rab 2453  df-v 2728

This theorem is referenced by:  notab  3392  intmin2  3850  euen1  6768  bj-omind  13816