Theorem rabab 2781

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 2481	. 2 |- { x e. _V | ph } = { x | ( x e. _V /\ ph ) }
2		vex 2763	. . . 4 |- x e. _V
3	2	biantrur 303	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	abbii 2309	. 2 |- { x | ph } = { x | ( x e. _V /\ ph ) }
5	1, 4	eqtr4i 2217	1 |- { x e. _V | ph } = { x | ph }

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2164   {cab 2179   {crab 2476   _Vcvv 2760

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-rab 2481  df-v 2762

This theorem is referenced by:  notab  3429  intmin2  3896  euen1  6856  bj-omind  15426