Theorem rabab 2575

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 2315	. 2 |- { x e. _V | ph } = { x | ( x e. _V /\ ph ) }
2		vex 2560	. . . 4 |- x e. _V
3	2	biantrur 287	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	abbii 2153	. 2 |- { x | ph } = { x | ( x e. _V /\ ph ) }
5	1, 4	eqtr4i 2063	1 |- { x e. _V | ph } = { x | ph }

Syntax hints:   /\ wa 97   = wceq 1243   e. wcel 1393   {cab 2026   {crab 2310   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-rab 2315  df-v 2559

This theorem is referenced by:  notab  3207  intmin2  3641  euen1  6282  bj-omind  10058