Theorem rabab 2707

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 2425	. 2 |- { x e. _V | ph } = { x | ( x e. _V /\ ph ) }
2		vex 2689	. . . 4 |- x e. _V
3	2	biantrur 301	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	abbii 2255	. 2 |- { x | ph } = { x | ( x e. _V /\ ph ) }
5	1, 4	eqtr4i 2163	1 |- { x e. _V | ph } = { x | ph }

Syntax hints:   /\ wa 103   = wceq 1331   e. wcel 1480   {cab 2125   {crab 2420   _Vcvv 2686

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-rab 2425  df-v 2688

This theorem is referenced by:  notab  3346  intmin2  3797  euen1  6696  bj-omind  13132