Theorem rabab 3223

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 2921	. 2 |- { x e. _V | ph } = { x | ( x e. _V /\ ph ) }
2		vex 3203	. . . 4 |- x e. _V
3	2	biantrur 527	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	abbii 2739	. 2 |- { x | ph } = { x | ( x e. _V /\ ph ) }
5	1, 4	eqtr4i 2647	1 |- { x e. _V | ph } = { x | ph }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rab 2921  df-v 3202

This theorem is referenced by:  notab  3897  intmin2  4504  euen1  8026  cardf2  8769  hsmex2  9255  imageval  32037  rmxyelqirr  37475  dfrcl2  37966